Q.Find the distance between the points $P(1, -3, 4)$ and $Q(-4, 1, 2)$.

Showing the 12 most recent of 18 on this concept.

1. TG EAPCET 2026Set eng-2026-05-09-FN1 markMCQ
   Q.The tangent drawn at a point P on the circle $x^2 + y^2 + 6x + 6y - 2 = 0$ cuts the line $5x - 2y + 6 = 0$ at a point Q. If PQ = 5, then a point Q having integral coordinates is (A) $(0, 3)$ (B) $(2, 8)$ (C) $(-2, -2)$ (D) $(-4, -7)$

   ›Reveal solutionSolution

   The key idea is to use the length of the tangent from an external point Q to the circle: $PQ^2 = \text{(power of Q)}$. We find the circle’s center and radius, set up the power condition, and test which given point satisfies $PQ = 5$.

   Concept & Intuition

   When a tangent is drawn from an external point Q to a circle, the segment from Q to the point of tangency P is perpendicular to the radius at P. A classic result (the tangent-secant power theorem) tells us that the square of the length of the tangent from Q equals the power of Q with respect to the circle:

   $PQ^2 = QO^2 - r^2$

   where O is the center and r the radius. So instead of finding P explicitly, we can directly test each candidate Q by checking whether its distance to the center, minus the square of the radius, equals $5^2 = 25$.

   Step-by-step solution

   1. Rewrite the circle equation in standard form

      Given: $x^2 + y^2 + 6x + 6y - 2 = 0$

      Complete the square:

      $(x^2 + 6x) + (y^2 + 6y) = 2$

      $(x^2 + 6x + 9) + (y^2 + 6y + 9) = 2 + 9 + 9$

      $(x + 3)^2 + (y + 3)^2 = 20$

      So center $O = (-3, -3)$ and radius $r = \sqrt{20} = 2\sqrt{5}$.

   2. Apply the tangent length formula

      For any point Q = $(x, y)$, the length of the tangent from Q to the circle is

   $PQ = \sqrt{ (x+3)^2 + (y+3)^2 - 20 }$

   We are told $PQ = 5$, so

   $(x+3)^2 + (y+3)^2 - 20 = 25$

   $(x+3)^2 + (y+3)^2 = 45$

   This is the condition Q must satisfy.

   3. Test each option

      • (A) $(0, 3)$: $(0+3)^2 + (3+3)^2 = 9 + 36 = 45$ ✓
      • (B) $(2, 8)$: $(2+3)^2 + (8+3)^2 = 25 + 121 = 146 \neq 45$ ✗
      • (C) $(-2, -2)$: $(-2+3)^2 + (-2+3)^2 = 1 + 1 = 2 \neq 45$ ✗
      • (D) $(-4, -7)$: $(-4+3)^2 + (-7+3)^2 = 1 + 16 = 17 \neq 45$ ✗

      Only option (A) satisfies the condition.

   Watch out

   A common mistake is to forget that the tangent length formula uses the square of the distance from Q to the center minus the square of the radius. Also, note that Q must lie on the given line $5x - 2y + 6 = 0$ — but here the problem only asks for a point Q with integral coordinates that satisfies $PQ = 5$, and the options are pre-selected. Option (A) indeed lies on that line: $5(0) - 2(3) + 6 = 0$ ✓.

   Tip

   You never need to find the actual point of tangency P. The power of a point is a powerful shortcut: it directly links the external point to the tangent length without solving for the contact point.

   ✓Final answer

   The correct option is (A).

   ANSWER: A

2. TG EAPCET 2026Set eng-2026-05-11-FN1 markMCQ
   Q.A straight line passes through a point A(2,5) and makes an angle of $45^\circ$ with the positive X-axis when measured in the positive direction. If this straight line intersects the line passing through the points (1,–2) and (3,–4) at B, then AB = (A) $2\sqrt{2}$ (B) $5\sqrt{2}$ (C) $4\sqrt{2}$ (D) $\sqrt{82}$

   ›Reveal solutionSolution

   The line through $A(2,5)$ with slope $\tan45^\circ=1$ is $y=x+3$; the line through $(1,-2),(3,-4)$ is $y=-x-1$. They meet at $B(-2,1)$, giving $AB=\sqrt{32}=4\sqrt2$ — option (C).

   Line through $A$. Angle $45^\circ$ with the positive X-axis gives slope $m_1=\tan45^\circ=1$. Through $A(2,5)$:

   $y-5=1\cdot(x-2)\;\Rightarrow\;y=x+3.$

   Line through the two given points. Slope

   $m_2=\frac{-4-(-2)}{3-1}=\frac{-2}{2}=-1.$

   Through $(1,-2)$:

   $y+2=-1\cdot(x-1)\;\Rightarrow\;y=-x-1.$

   Intersection $B$. Set $x+3=-x-1$:

   $2x=-4\;\Rightarrow\;x=-2,\qquad y=x+3=1,$

   so $B=(-2,1)$.

   Distance $AB$.

   $AB=\sqrt{(2-(-2))^2+(5-1)^2}=\sqrt{4^2+4^2}=\sqrt{32}=4\sqrt2.$

   ✓Final answer

   $AB=4\sqrt2$ — option (C).

3. TG EAPCET 2025Set eng-2025-05-02-FN1 markMCQ
   Q.A variable straight-line L with negative slope passes through the point (4,9) and cuts the positive coordinate axes in A and B. If O is the origin, then the minimum value of OA + OB is (A) 25 (B) 12 (C) 13 (D) 5

   ›Reveal solutionSolution

   The problem reduces to minimizing the sum of intercepts of a line with fixed point (4,9) and negative slope. Using the intercept form and AM–GM inequality, the minimum sum is 25, so the correct option is (A).

   We have a line with negative slope passing through (4,9) that meets the positive x‑axis at A and the positive y‑axis at B. We want the smallest possible value of OA + OB, where OA is the x‑intercept and OB is the y‑intercept.

   Concept & Intuition

   For a line with intercepts $a$ (on x‑axis) and $b$ (on y‑axis), its equation is $\frac{x}{a} + \frac{y}{b} = 1$. Since the line goes through (4,9), we have $\frac{4}{a} + \frac{9}{b} = 1$. The sum we want is $S = a + b$. Because the slope is negative, both $a$ and $b$ are positive. The constraint links $a$ and $b$ in a way that lets us use the AM–GM inequality to find the minimum of $a+b$.

   ---

   Step‑by‑step solution

   1. Set up the intercept form Let the x‑intercept be $a > 0$ and the y‑intercept be $b > 0$. The line equation is

   $\frac{x}{a} + \frac{y}{b} = 1.$

   Since the point $(4,9)$ lies on the line,

   $\frac{4}{a} + \frac{9}{b} = 1. \qquad(1)$

   2. Express the sum to minimize

      We need the minimum of $S = a + b$ under condition (1).

   3. Apply AM–GM inequality

      A classic trick: rewrite (1) as

   $\frac{4}{a} + \frac{9}{b} = 1.$

   Multiply both sides by $a+b$? Better: use the inequality

   $(a+b)\left(\frac{4}{a} + \frac{9}{b}\right) \ge (\sqrt{4} + \sqrt{9})^2 = (2+3)^2 = 25.$

   This is a direct application of the Cauchy–Schwarz or AM–GM form: for positive numbers,

   $(x+y)\left(\frac{p}{x} + \frac{q}{y}\right) \ge (\sqrt{p} + \sqrt{q})^2.$

   Here $x = a$, $y = b$, $p = 4$, $q = 9$.

   4. Use the constraint From (1), $\frac{4}{a} + \frac{9}{b} = 1$, so

   $(a+b) \cdot 1 \ge 25 \quad \Rightarrow \quad a+b \ge 25.$

   Hence the minimum possible value of $OA + OB$ is at least 25.

   5. Check when equality occurs Equality in the AM–GM / Cauchy–Schwarz form holds when

   $\frac{a}{4} = \frac{b}{9} \quad \text{(more precisely, } \frac{a}{\sqrt{4}} = \frac{b}{\sqrt{9}} \text{)}.$

   So $\frac{a}{2} = \frac{b}{3}$ ⇒ $b = \frac{3}{2}a$.

   Substitute into (1):

   $\frac{4}{a} + \frac{9}{\frac{3}{2}a} = \frac{4}{a} + \frac{6}{a} = \frac{10}{a} = 1 \quad \Rightarrow \quad a = 10.$

   Then $b = 15$. Both are positive, slope is $-\frac{b}{a} = -1.5$ (negative), so the line indeed cuts the positive axes. The sum is $10+15=25$.

   Tip

   The inequality $(a+b)\left(\frac{4}{a}+\frac{9}{b}\right) \ge (\sqrt{4}+\sqrt{9})^2$ is a special case of Titu’s lemma or Cauchy–Schwarz. It’s the fastest route when the constraint is a sum of reciprocals.

   Watch out

   A common mistake is to set $a = b$ by symmetry — but the point (4,9) is not symmetric, so that guess gives a larger sum. Always check the equality condition.

   ---

   ✓Final answer

   The correct option is (A).

   ANSWER: A

4. TG EAPCET 2025Set eng-2025-05-02-FN1 markMCQ
   Q.The focal distance of a point (5,5) on the parabola $x^2 - 2x - 4y + 5 = 0$ is (A) $5$ (B) $8$ (C) $10$ (D) $12$

   ›Reveal solutionSolution

   The focal distance is the distance from a point on a parabola to its focus. For the given parabola, after rewriting in standard form, the focus is at $(1,2)$, so the distance from $(5,5)$ to $(1,2)$ is $5$. The correct option is (A).

   ---

   Concept & Intuition

   The focal distance of a point on a parabola is simply the distance from that point to the focus. For a parabola defined by $x^2 = 4ay$ (vertex at origin, opening upward), the focus is at $(0,a)$. Here the equation is shifted, so we first rewrite it in vertex form to identify the focus. Then we compute the Euclidean distance.

   ---

   Step-by-step solution

   1. Rewrite the given equation

      The equation is $x^2 - 2x - 4y + 5 = 0$.

      Complete the square for the $x$-terms:

      $x^2 - 2x = (x-1)^2 - 1$.

      Substitute:

      $(x-1)^2 - 1 - 4y + 5 = 0$

      $\Rightarrow (x-1)^2 - 4y + 4 = 0$

      $\Rightarrow (x-1)^2 = 4y - 4$

      $\Rightarrow (x-1)^2 = 4(y-1)$.

   2. Identify the standard form and parameters

      The equation $(x-1)^2 = 4(y-1)$ is of the form $(x-h)^2 = 4a(y-k)$, where $(h,k)$ is the vertex.

      Here $h=1$, $k=1$, and $4a = 4 \Rightarrow a = 1$.

      The parabola opens upward (since $a>0$).

      For this standard form, the focus is at $(h, k+a) = (1, 1+1) = (1,2)$.

   3. Compute the focal distance

      The given point is $P = (5,5)$.

      Distance from $P$ to focus $F = (1,2)$:

   $\text{Distance} = \sqrt{(5-1)^2 + (5-2)^2} = \sqrt{4^2 + 3^2} = \sqrt{16+9} = \sqrt{25} = 5.$

   Tip

   The focal distance is not the same as the distance from the point to the directrix — that would be equal only if the point lies on the parabola (which it does), but here we directly compute the distance to the focus.

   4. Match with options The value $5$ corresponds to option (A).

   ---

   ✓Final answer

   The correct option is (A).

   ANSWER: A

5. TG EAPCET 2025Set eng-2025-05-02-AN1 markMCQ
   Q.If the points $(1,1,\lambda)$ and $(-3,0,1)$ are equidistant from the plane $3x+4y-12z+13=0$, then the values of $\lambda$ are (A) $-1, \dfrac{7}{3}$ (B) $1, -\dfrac{7}{3}$ (C) $-1, -\dfrac{7}{3}$ (D) $1, \dfrac{7}{3}$

   ›Reveal solutionSolution

   Setting the two perpendicular distances equal gives $|20-12\lambda|=8$, whose solutions are $\lambda=1$ and $\lambda=\tfrac{7}{3}$ — option (D).

   Distance formula. The distance from $(x_0,y_0,z_0)$ to $3x+4y-12z+13=0$ is

   $\frac{|3x_0+4y_0-12z_0+13|}{\sqrt{3^2+4^2+12^2}}=\frac{|3x_0+4y_0-12z_0+13|}{13}.$

   Evaluate at each point.

   $\text{At }(1,1,\lambda):\quad 3+4-12\lambda+13=20-12\lambda.$

   $\text{At }(-3,0,1):\quad -9+0-12+13=-8.$

   Equidistant condition (equal denominators):

   $|20-12\lambda|=|-8|=8.$

   Solve.

   $20-12\lambda=8\Rightarrow\lambda=1,\qquad 20-12\lambda=-8\Rightarrow 12\lambda=28\Rightarrow\lambda=\frac{7}{3}.$

   ✓Final answer

   The values are $\lambda=1$ and $\lambda=\dfrac{7}{3}$ — option (D).

6. TG EAPCET 2025Set eng-2025-05-03-FN1 markMCQ
   Q.If the point P$(x_1, y_1)$ lying on the curve $y = x^2 - x + 1$ is the closest point to the line $y = x - 3$ then the perpendicular distance from P to the line $3x + 4y - 2 = 0$ is (A) $1$ (B) $\dfrac{7}{5}$ (C) $\dfrac{16}{5}$ (D) $4$

   ›Reveal solutionSolution

   The closest point on a curve to a line is where the tangent is parallel to the line; solving gives P(1,1), and its distance to the given line is 1, so option (A).

   We need the point on $y = x^2 - x + 1$ that is closest to the line $y = x - 3$. The key idea: the shortest distance from a curve to a line occurs at a point where the tangent to the curve is parallel to the line. Why? Because if you imagine sliding a line parallel to the given one until it just touches the curve, the point of tangency is the closest point. This is a standard optimization trick — it avoids calculus with distances directly.

   1. Find the slope of the given line.

      The line is $y = x - 3$, so its slope is $1$.

   2. Find the slope of the tangent to the curve.

      The curve is $y = x^2 - x + 1$. Differentiate:

   $\frac{dy}{dx} = 2x - 1.$

   3. Set the tangent slope equal to the line’s slope. For the closest point,

   $2x - 1 = 1 \quad \Rightarrow \quad 2x = 2 \quad \Rightarrow \quad x = 1.$

   4. Find the corresponding y-coordinate. Substitute $x = 1$ into the curve:

   $y = 1^2 - 1 + 1 = 1.$

   So the point is $P(1, 1)$.

   Tip

   Always check that this point actually lies on the curve — it does. Also, the line $y = x - 3$ does not intersect the curve (try solving $x^2 - x + 1 = x - 3$ gives $x^2 - 2x + 4 = 0$, no real roots), so the closest point is indeed a tangency point, not an intersection.

   5. Now find the perpendicular distance from P to the line $3x + 4y - 2 = 0$. The distance formula from point $(x_1, y_1)$ to line $Ax + By + C = 0$ is

   $d = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}}.$

   Here $A = 3$, $B = 4$, $C = -2$, and $P = (1, 1)$:

   $d = \frac{|3(1) + 4(1) - 2|}{\sqrt{3^2 + 4^2}} = \frac{|3 + 4 - 2|}{\sqrt{9 + 16}} = \frac{|5|}{\sqrt{25}} = \frac{5}{5} = 1.$

   Watch out

   A common mistake is to use the line $y = x - 3$ in the distance formula instead of the given line $3x + 4y - 2 = 0$. Read carefully: the first line is only used to find P; the distance is asked from the second line.

   Thus the perpendicular distance is $1$.

   ✓Final answer

   The correct option is (A).

   ANSWER: A

7. TG EAPCET 2025Set eng-2025-05-03-FN1 markMCQ
   Q.If a line L passing through the point $A(-2, 4)$ makes an angle of $60^\circ$ with the positive direction of X-axis in anti-clockwise direction and $B(p, q)$ lying in the 3rd quadrant is a point on L at the distance of 6 units from the point A, then $\sqrt{p^2 + q^2 - 8q} =$ (A) $8$ (B) $7$ (C) $9$ (D) $6$

   ›Reveal solutionSolution

   The line through A at 60° has slope √3; using parametric form, point B is 6 units away in the 3rd quadrant; substituting into the expression gives 8.

   We are given a point $A(-2,4)$ and a line $L$ through it making a $60^\circ$ angle with the positive X-axis (counterclockwise). That means the slope is $\tan 60^\circ = \sqrt{3}$. The line also passes through a point $B(p,q)$ in the 3rd quadrant (so $p<0, q<0$), exactly 6 units from $A$. We need $\sqrt{p^2 + q^2 - 8q}$.

   Concept & Intuition

   When a line’s direction is known, the easiest way to locate a point at a given distance along it is to use the parametric form:

   $(x, y) = (x_0 + r\cos\theta,\; y_0 + r\sin\theta)$

   where $r$ is the signed distance from $(x_0, y_0)$. Here $\theta = 60^\circ$, so $\cos 60^\circ = \frac12$, $\sin 60^\circ = \frac{\sqrt{3}}{2}$. The distance is 6, but the sign of $r$ determines which side of A we go. Since B is in the 3rd quadrant (both coordinates negative), we must go in the direction that makes both coordinates decrease from $A(-2,4)$. That means moving opposite to the positive direction of the line — so we take $r = -6$.

   Step-by-step

   1. Parametric coordinates of B Starting from $A(-2,4)$, with $r = -6$:

   $p = -2 + (-6)\cos 60^\circ = -2 - 6\cdot\frac12 = -2 - 3 = -5$

   $q = 4 + (-6)\sin 60^\circ = 4 - 6\cdot\frac{\sqrt{3}}{2} = 4 - 3\sqrt{3}$

   2. Check quadrant

      $p = -5 < 0$.

      $q = 4 - 3\sqrt{3} \approx 4 - 5.196 = -1.196 < 0$.

      So B is indeed in the 3rd quadrant. Good.

   3. Compute the expression

      We need $\sqrt{p^2 + q^2 - 8q}$.

      First, $p^2 = (-5)^2 = 25$.

      Next, $q^2 = (4 - 3\sqrt{3})^2 = 16 - 24\sqrt{3} + 27 = 43 - 24\sqrt{3}$.

      Then $-8q = -8(4 - 3\sqrt{3}) = -32 + 24\sqrt{3}$.

      Add them:

   $p^2 + q^2 - 8q = 25 + (43 - 24\sqrt{3}) + (-32 + 24\sqrt{3})$

   The $\sqrt{3}$ terms cancel: $-24\sqrt{3} + 24\sqrt{3} = 0$.

   Constant terms: $25 + 43 - 32 = 36$.

   So the expression becomes $\sqrt{36} = 6$.

   Tip

   The cancellation of the irrational part is not a coincidence — it happens because the expression was designed to simplify nicely when B lies on that specific line.

   Watch out

   A common mistake is to take $r = +6$ (going in the direction of increasing x and y). That would give $p = -2 + 3 = 1$ (positive), which is not in the 3rd quadrant. Always check the quadrant condition.

   ✓Final answer

   The correct option is (D).

   ANSWER: D

8. TG EAPCET 2024Set eng-2024-05-09-AN1 markMCQ
   Q.$L = x \cos \alpha + y \sin \alpha - p = 0$ represents a line perpendicular to the line $x + y + 1 = 0$. If $p$ is positive, $\alpha$ lies in the fourth quadrant and perpendicular distance from $(\sqrt{2}, \sqrt{2})$ to the line $L = 0$ is $5$ units then $p =$ (A) $5$ (B) $\dfrac{5}{2}$ (C) $10$ (D) $\dfrac{15}{2}$

   ›Reveal solutionSolution

   The line $L$ is perpendicular to $x+y+1=0$, so its normal vector is parallel to $(1,1)$. Using the given distance from $(\sqrt2,\sqrt2)$ and the quadrant condition for $\alpha$, we find $p = 5$, which corresponds to option (A).

   ---

   We are told that $L = x\cos\alpha + y\sin\alpha - p = 0$ is a line, and that it is perpendicular to the line $x + y + 1 = 0$.

   The key idea: the normal vector of $L$ is $(\cos\alpha,\sin\alpha)$, and the normal vector of $x+y+1=0$ is $(1,1)$. The key idea: the normal vector of $L$ is $(\cos\alpha,\sin\alpha)$, and the normal vector of $x+y+1=0$ is $(1,1)$. Two lines are perpendicular exactly when their normal vectors are perpendicular — if $\mathbf{d}_1,\mathbf{d}_2$ are the direction vectors and $\mathbf{n}_1,\mathbf{n}_2$ the normals of the two lines, then $\mathbf{d}_1\cdot\mathbf{n}_1=0$ and $\mathbf{d}_2\cdot\mathbf{n}_2=0$; the lines being perpendicular means $\mathbf{d}_1\cdot\mathbf{d}_2=0$, which in turn forces $\mathbf{n}_1\cdot\mathbf{n}_2=0$. (Quick check: $L_1:x=0$ has normal $(1,0)$ and $L_2:y=0$ has normal $(0,1)$ — both the lines and their normals are perpendicular.)

   Thus for our lines:

   • Normal of $L$: $(\cos\alpha,\sin\alpha)$
   • Normal of $x+y+1=0$: $(1,1)$

   Perpendicular condition:

   $(\cos\alpha,\sin\alpha) \cdot (1,1) = 0 \quad\Rightarrow\quad \cos\alpha + \sin\alpha = 0$

   So $\sin\alpha = -\cos\alpha$, i.e. $\tan\alpha = -1$.

   Now $\alpha$ lies in the fourth quadrant. In the fourth quadrant, $\cos\alpha > 0$, $\sin\alpha < 0$, and $\tan\alpha = -1$ gives $\alpha = -\frac{\pi}{4}$ (or $315^\circ$). So:

   $\cos\alpha = \frac{1}{\sqrt{2}},\quad \sin\alpha = -\frac{1}{\sqrt{2}}$

   Thus the line $L$ becomes:

   $x\cdot\frac{1}{\sqrt{2}} + y\cdot\left(-\frac{1}{\sqrt{2}}\right) - p = 0$

   Multiply through by $\sqrt{2}$:

   $x - y - p\sqrt{2} = 0$

   Now we are told the perpendicular distance from $(\sqrt{2},\sqrt{2})$ to this line is $5$ units. The distance from a point $(x_1,y_1)$ to line $ax+by+c=0$ is:

   $\frac{|ax_1+by_1+c|}{\sqrt{a^2+b^2}}$

   Here $a=1$, $b=-1$, $c = -p\sqrt{2}$. So:

   $\text{Distance} = \frac{|1\cdot\sqrt{2} + (-1)\cdot\sqrt{2} - p\sqrt{2}|}{\sqrt{1^2+(-1)^2}} = \frac{|\sqrt{2} - \sqrt{2} - p\sqrt{2}|}{\sqrt{2}} = \frac{|-p\sqrt{2}|}{\sqrt{2}} = |p|$

   Given this distance is $5$, we have $|p| = 5$. Since $p$ is positive, $p = 5$.

   Watch out

   A common mistake is to forget that the perpendicular condition involves normals, not direction vectors, leading to $\cos\alpha = \sin\alpha$ instead of $\cos\alpha + \sin\alpha = 0$. Always check: normals perpendicular $\iff$ lines perpendicular.

   Tip

   Notice how the distance simplified beautifully to $|p|$ — this is because the point $(\sqrt2,\sqrt2)$ lies exactly on the line $x-y=0$, which is parallel to our line’s direction, making the algebra collapse.

   ✓Final answer

   The correct option is (A).

   ANSWER: A

9. TG EAPCET 2024Set eng-2024-05-10-FN1 markMCQ
   Q.A plane $\pi_1$ passing through the point $3\vec{i} - 7\vec{j} + 5\vec{k}$ is perpendicular to the vector $\vec{i} + 2\vec{j} - 2\vec{k}$ and another plane $\pi_2$ passing through the point $2\vec{i} + 7\vec{j} - 8\vec{k}$ is perpendicular to the vector $3\vec{i} + 2\vec{j} + 6\vec{k}$. If $p_1$ and $p_2$ are the perpendicular distances from the origin to the planes $\pi_1$ and $\pi_2$ respectively, then $p_1 - p_2 =$ (A) $1$ (B) $2$ (C) $3$ (D) $4$

   ›Reveal solutionSolution

   The key idea is to write each plane in normal form using a point and a normal vector, then compute the perpendicular distance from the origin using the formula $p = \frac{|\vec{a} \cdot \vec{n}|}{|\vec{n}|}$. The difference $p_1 - p_2$ simplifies to $2$, so the correct option is (B).

   Concept and Intuition

   A plane can be defined by a point on it and a normal vector. The distance from the origin to a plane is the absolute value of the scalar projection of any point’s position vector onto the unit normal. Since the normal vectors are given, we can directly compute the distances without finding the full Cartesian equations.

   Step-by-step solution

   1. Equation of plane $\pi_1$ The plane passes through $A(3, -7, 5)$ and has normal $\vec{n}_1 = \vec{i} + 2\vec{j} - 2\vec{k}$. The equation is $\vec{r} \cdot \vec{n}_1 = \vec{a} \cdot \vec{n}_1$, where $\vec{a} = 3\vec{i} - 7\vec{j} + 5\vec{k}$. Compute:

   $\vec{a} \cdot \vec{n}_1 = 3(1) + (-7)(2) + 5(-2) = 3 - 14 - 10 = -21.$

   So $\pi_1: \vec{r} \cdot (\vec{i} + 2\vec{j} - 2\vec{k}) = -21$.

   2. Distance $p_1$ from origin to $\pi_1$ The distance from origin $(0,0,0)$ to plane $\vec{r} \cdot \vec{n} = d$ is $\frac{|d|}{|\vec{n}|}$. Here $d = -21$, $|\vec{n}_1| = \sqrt{1^2 + 2^2 + (-2)^2} = \sqrt{9} = 3$. Hence

   $p_1 = \frac{|-21|}{3} = \frac{21}{3} = 7.$

   3. Equation of plane $\pi_2$ The plane passes through $B(2, 7, -8)$ and has normal $\vec{n}_2 = 3\vec{i} + 2\vec{j} + 6\vec{k}$. Compute $\vec{b} \cdot \vec{n}_2$:

   $2(3) + 7(2) + (-8)(6) = 6 + 14 - 48 = -28.$

   So $\pi_2: \vec{r} \cdot (3\vec{i} + 2\vec{j} + 6\vec{k}) = -28$.

   4. Distance $p_2$ from origin to $\pi_2$ $|\vec{n}_2| = \sqrt{3^2 + 2^2 + 6^2} = \sqrt{9 + 4 + 36} = \sqrt{49} = 7$. Thus

   $p_2 = \frac{|-28|}{7} = \frac{28}{7} = 4.$

   5. Compute $p_1 - p_2$

   $p_1 - p_2 = 7 - 4 = 3.$

   Watch out

   A common mistake is to forget the absolute value in the distance formula, which would give negative distances. Here both $d$ values are negative, but distances are positive.

   Tip

   Notice that the constant term $d$ is simply the dot product of the given point with the normal. No need to expand the plane equation fully.

   ✓Final answer

   The correct option is (C).

   ANSWER: C

10. TG EAPCET 2023Set eng-2023-05-13-FN1 markMCQ
    Q.If $L_1$ is a line passing through the point $P(4, -3)$ and perpendicular to the line $3x - 4y + k = 0$, then the distance of $P$ from the line $5x - 3y - 2 = 0$ measured along the line $L_1$ is (A) $5$ (B) $\sqrt{13}$ (C) $\sqrt{41}$ (D) $13$

    ›Reveal solutionSolution

    The key idea is to find the intersection of the given line with the line through P perpendicular to the first line, then compute the distance between P and that intersection. The answer is 5.

    We are asked: the distance of P from the line $5x - 3y - 2 = 0$ measured along the line $L_1$.

    This means: start at P, travel along $L_1$ until you hit the line $5x - 3y - 2 = 0$; the distance you travel is what we want. So we need the intersection point of $L_1$ with that line, then the distance from P to that point.

    ---

    1. Find the equation of $L_1$.

    $L_1$ is perpendicular to $3x - 4y + k = 0$. The slope of that line is $\frac{3}{4}$ (rewrite as $y = \frac{3}{4}x + \frac{k}{4}$).

    A line perpendicular to it has slope $-\frac{4}{3}$ (negative reciprocal).

    $L_1$ passes through $P(4, -3)$, so its equation:

    $y + 3 = -\frac{4}{3}(x - 4)$

    Multiply: $3y + 9 = -4x + 16$

    So $4x + 3y - 7 = 0$.

    Tip

    The constant $k$ in the original line doesn't affect the slope, so it doesn't affect $L_1$'s direction — only its position. That's why we could find $L_1$ without knowing $k$.

    2. Find where $L_1$ meets the line $5x - 3y - 2 = 0$.

    Solve the system:

    $\begin{cases} 4x + 3y = 7 \\ 5x - 3y = 2 \end{cases}$

    Add the equations: $9x = 9 \Rightarrow x = 1$.

    Substitute into $4(1) + 3y = 7 \Rightarrow 3y = 3 \Rightarrow y = 1$.

    So the intersection point is $Q(1, 1)$.

    3. Compute the distance from $P(4, -3)$ to $Q(1, 1)$.

    Distance formula:

    $\sqrt{(4-1)^2 + (-3-1)^2} = \sqrt{3^2 + (-4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5.$

    Watch out

    A common mistake is to compute the perpendicular distance from P to the line $5x - 3y - 2 = 0$ directly. That would give $\frac{|5(4) - 3(-3) - 2|}{\sqrt{5^2 + (-3)^2}} = \frac{27}{\sqrt{34}}$, which is not among the options. The phrase "measured along $L_1$" means we travel along $L_1$, not along the perpendicular.

    ✓Final answer

    The correct option is (A).

    ANSWER: A

11. TG EAPCET 2023Set eng-2023-05-13-FN1 markMCQ
    Q.Let $A\ (4, 3, 5),\ B\ (1, -2, 1),\ C\ (3, 2, 1)$ be the vertices of a triangle $ABC$. If the internal bisector of $\angle BAC$ meet the side $BC$ at $D$, then $CD =$ (A) $\dfrac{\sqrt{5}}{4}$ (B) $\dfrac{3\sqrt{5}}{4}$ (C) $2\sqrt{5}$ (D) $\dfrac{5\sqrt{5}}{2}$

    ›Reveal solutionSolution

    The internal angle bisector theorem gives the ratio in which D divides BC as AB:AC. After computing side lengths, CD is a fraction of BC; the result is $\frac{3\sqrt{5}}{4}$, so option (B).

    We have triangle $ABC$ with vertices $A(4,3,5)$, $B(1,-2,1)$, $C(3,2,1)$. The internal bisector of $\angle BAC$ meets $BC$ at $D$. We need $CD$.

    Concept and intuition

    The key idea is the Angle Bisector Theorem: In any triangle, the internal bisector of an angle divides the opposite side into segments proportional to the adjacent sides.

    So for $\angle BAC$, the point $D$ on $BC$ satisfies

    $\frac{BD}{DC} = \frac{AB}{AC}.$

    Thus, if we compute the lengths $AB$ and $AC$, we can find the ratio $BD:DC$, and then $CD$ as a fraction of the whole side $BC$.

    Step-by-step solution

    1. Compute $AB$ $A(4,3,5)$, $B(1,-2,1)$

    $AB = \sqrt{(4-1)^2 + (3-(-2))^2 + (5-1)^2} = \sqrt{3^2 + 5^2 + 4^2} = \sqrt{9 + 25 + 16} = \sqrt{50} = 5\sqrt{2}.$

    2. Compute $AC$ $A(4,3,5)$, $C(3,2,1)$

    $AC = \sqrt{(4-3)^2 + (3-2)^2 + (5-1)^2} = \sqrt{1^2 + 1^2 + 4^2} = \sqrt{1 + 1 + 16} = \sqrt{18} = 3\sqrt{2}.$

    3. Apply the Angle Bisector Theorem

    $\frac{BD}{DC} = \frac{AB}{AC} = \frac{5\sqrt{2}}{3\sqrt{2}} = \frac{5}{3}.$

    So $BD : DC = 5 : 3$.

    This means $BC$ is divided into $5+3=8$ equal parts, and $DC$ corresponds to $3$ of those parts.

    4. Compute $BC$ $B(1,-2,1)$, $C(3,2,1)$

    $BC = \sqrt{(3-1)^2 + (2-(-2))^2 + (1-1)^2} = \sqrt{2^2 + 4^2 + 0^2} = \sqrt{4 + 16} = \sqrt{20} = 2\sqrt{5}.$

    5. Find $CD$ Since $DC = \frac{3}{8} \times BC$,

    $CD = \frac{3}{8} \cdot 2\sqrt{5} = \frac{6\sqrt{5}}{8} = \frac{3\sqrt{5}}{4}.$

    Tip

    Notice that $AB$ and $AC$ both contain $\sqrt{2}$, which cancels in the ratio — a common simplification that avoids messy arithmetic.

    Watch out

    A classic mistake is to invert the ratio: the bisector of $\angle A$ gives $BD/DC = AB/AC$, not $AC/AB$. Double-check which side corresponds to which segment.

    ✓Final answer

    The correct option is (B).

    ANSWER: B

12. TG EAPCET 2023Set eng-2023-05-12-FN1 markMCQ
    Q.If the slope of a straight line passing through $A(3,2)$ is $3/4$, then the coordinates of the two points on the same line that are $5$ units away from $A$ are (A) $(-7,5), (1,-1)$ (B) $(7,5), (-1,-1)$ (C) $(6,9), (-2,3)$ (D) $(6,3), (-2,-3)$

    ›Reveal solutionSolution

    Use the parametric form of a line from a given point with a known slope to find points at a specific distance. The two points are $(7,5)$ and $(-1,-1)$, so the correct option is (B).

    We have point $A(3,2)$ and slope $m = \frac{3}{4}$. We want the two points on this line that are exactly $5$ units away from $A$. The key idea: a line with a given slope has a direction vector; moving along that vector by a certain distance gives the required points.

    1. Direction vector from the slope

       Slope $\frac{3}{4}$ means that for every $4$ units moved horizontally, we move $3$ units vertically. So a direction vector is $(4,3)$. Its length is $\sqrt{4^2 + 3^2} = \sqrt{16+9} = \sqrt{25} = 5$.

       This is perfect: the vector itself has length exactly $5$.

    2. Parametric form of the line

       Any point on the line through $A$ can be written as

    $(x, y) = (3,2) + t \cdot (4,3)$

    where $t$ is a real number. When $t=0$, we are at $A$. When $t=1$, we move exactly one full direction vector — that is, $5$ units — to $(7,5)$. When $t=-1$, we move in the opposite direction $5$ units to $(-1,-1)$.

    3. Check the distance Distance from $A$ to $(7,5)$:

    $\sqrt{(7-3)^2 + (5-2)^2} = \sqrt{4^2 + 3^2} = \sqrt{25} = 5.$

    Distance from $A$ to $(-1,-1)$:

    $\sqrt{(-1-3)^2 + (-1-2)^2} = \sqrt{(-4)^2 + (-3)^2} = \sqrt{25} = 5.$

    Both work.

    4. Match with options The points are $(7,5)$ and $(-1,-1)$. Looking at the choices:
       • (A) $(-7,5), (1,-1)$ — no
       • (B) $(7,5), (-1,-1)$ — yes
       • (C) $(6,9), (-2,3)$ — no
       • (D) $(6,3), (-2,-3)$ — no

    Watch out

    A common mistake is to forget the two directions: moving forward and backward along the line. Only one point is found if you only consider $t=1$.

    Tip

    When the direction vector’s length already equals the desired distance, the problem simplifies to just adding and subtracting that vector — no scaling needed.

    ✓Final answer

    The correct option is (B).

    ANSWER: B