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Mathematics · Class 11 Science

Ch 6Two Dimensional Analytical Geometry — Class 11 Mathematics, concept-first.

Analytical (coordinate) geometry studies geometric figures using algebra, by fixing a coordinate system and translating geometric conditions into equations. The idea was born in the 1630s, when two French mathematician-philosophers, René Descartes (1596–1650) and Pierre de Fermat, independently founded the subject by a…

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Locus of a Point

A point is not a thing but a place — a position located, with reference to a coordinate system, by a unique ordered pair : is the (signed) distance from the -axis and the (signed) distance from the -axis.

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Chapter contents

The NCERT structure, section by section. Open a section to see its questions, then read the concept-first solution.

Introduction

Analytical (coordinate) geometry studies geometric figures using algebra, by fixing a coordinate system and translating geometric conditions into equations.

6.2

Locus of a Point

A point is not a physical object but a position — indicated on paper by a dot, and located algebraically (once a coordinate system is fixed) by a unique ordered pair of real numbers .

+Exercise 6.1i15 questions
  1. Q1Find the locus of $P$, if for all values of $\alpha$, the co-ordinates of a moving point $P$ is (i) $(9\cos\alpha,\ 9\sin\alpha)$ (ii) $(9\c…Free
  2. Q2Find the locus of a point $P$ that moves at a constant distance of (i) two units from the $x$-axis (ii) three units from the $y$-axis.Free
  3. Q3If $\theta$ is a parameter, find the equation of the locus of a moving point, whose coordinates are $x = a\cos^3\theta,\ y = a\sin^3\theta$.Free
  4. Q4Find the value of $k$ and $b$, if the points $P(-3, 1)$ and $Q(2, b)$ lie on the locus of $x^2 - 5x + ky = 0$.Preview
  5. Q5A straight rod of length $8$ units slides with its ends $A$ and $B$ always on the $x$ and $y$ axes respectively. Find the locus of the mid p…Preview
  6. Q6Find the equation of the locus of a point such that the sum of the squares of the distances from the points $(3, 5)$, $(1, -1)$ is equal to…Preview
  7. Q7Find the equation of the locus of the point $P$ such that the line segment $AB$, joining the points $A(1, -6)$ and $B(4, -2)$, subtends a ri…Preview
  8. Q8If $O$ is the origin and $R$ is a variable point on $y^2 = 4x$, then find the equation of the locus of the mid-point of the line segment $OR…Preview
  9. Q9The coordinates of a moving point $P$ are $\left(\dfrac{a}{2}\left(\csc\theta + \sin\theta\right),\ \dfrac{b}{2}\left(\csc\theta - \sin\thet…Preview
  10. Q10If $P(2, -7)$ is a given point and $Q$ is a point on $2x^2 + 9y^2 = 18$, then find the equation of the locus of the mid-point of $PQ$.Preview
  11. Q11If $R$ is any point on the $x$-axis and $Q$ is any point on the $y$-axis, and $P$ is a variable point on $RQ$ with $RP = b$, $PQ = a$, then…Preview
  12. Q12If the points $P(6, 2)$ and $Q(-2, 1)$, and $R$, are the vertices of a $\triangle PQR$, and $R$ is a point on the locus $y = x^2 - 3x + 4$,…Preview
  13. Q13If $Q$ is a point on the locus of $x^2 + y^2 + 4x - 3y + 7 = 0$, then find the equation of the locus of $P$ which divides the segment $OQ$ e…Preview
  14. Q14Find the points on the locus of points that are $3$ units from the $x$-axis and $5$ units from the point $(5, 1)$.Preview
  15. Q15The sum of the distances of a moving point from the points $(4, 0)$ and $(-4, 0)$ is always $10$ units. Find the equation of the locus of th…Preview
6.3

Straight Lines

Because rewriting (dividing throughout by whichever of is nonzero) always removes one constant, a straight line's equation genuinely contains only two independent arbitrary constants — so two pieces o…

6.3.1

The Relationship Between the Angle of Inclination and Slope

Angle of inclination. For a straight line, the angle of inclination is the angle the line makes with the -axis, measured in the counter-clockwise (positive) direction from the positive -axis.

6.3.2

Intercepts of a Line

The intercepts of a line are the points at which it crosses the coordinate axes. The -intercept is the point where the line meets the -axis — where the -coordinate is zero — and the -intercept is wher…

6.3.3

Different Forms of an Equation of a Straight Line

Given any two independent pieces of information about a line from among points, slope, and intercepts, its equation can be written directly.

+Exercise 6.2i15 questions
  1. Q1Find the equation of the lines passing through the point $(1, 1)$ (i) with $y$-intercept $(-4)$ (ii) with slope $3$ (iii) and $(-2, 3)$ (iv)…Free
  2. Q2If $P(r, c)$ is the mid point of a line segment between the axes, then show that $\dfrac{x}{r} + \dfrac{y}{c} = 2$.Free
  3. Q3Find the equation of the line passing through the point $(1, 5)$ and also dividing the co-ordinate axes in the ratio $3:10$.Free
  4. Q4If $p$ is the length of the perpendicular from the origin to the line whose intercepts on the axes are $a$ and $b$, then show that $\dfrac{1…Preview
  5. Q5The normal boiling point of water is $100^\circ$C or $212^\circ$F, and the freezing point of water is $0^\circ$C or $32^\circ$F. (i) Find th…Preview
  6. Q6An object was launched from a place $P$ at a constant speed to hit a target. At the $15$th second it was $1400$ m away from the target, and…Preview
  7. Q7The population of a city in the years $2005$ and $2010$ are $1{,}35{,}000$ and $1{,}45{,}000$ respectively. Find the approximate population…Preview
  8. Q8Find the equation of the line, if the perpendicular drawn from the origin makes an angle $30^\circ$ with the $x$-axis and its length is $12$…Preview
  9. Q9Find the equation of the straight lines passing through $(8, 3)$ and having intercepts on the axes whose sum is $1$.Preview
  10. Q10Show that the points $(1, 3)$, $(2, 1)$ and $\left(\dfrac12, 4\right)$ are collinear, by using (i) the concept of slope, (ii) the equation o…Preview
  11. Q11A straight line is passing through the point $A(1, 2)$ with slope $\dfrac{5}{12}$. Find the points on the line which are $13$ units away fro…Preview
  12. Q12A $150$ m long train is moving with constant velocity of $12.5$ m/s. Find (i) the equation of the motion of the train (ii) the time taken to…Preview
  13. Q13A spring was hung from a hook in the ceiling. A number of different weights were attached to the spring to make it stretch, and the total le…Preview
  14. Q14A family is using Liquefied Petroleum Gas (LPG) of weight $14.2$ kg for consumption (the full cylinder weighs $29.5$ kg, which includes the…Preview
  15. Q15In a shopping mall there is a hall of cuboid shape with dimensions $800 \times 800 \times 720$ units, which needs to have the facility of an…Preview
6.3.4

General Form to Other Forms

Every general linear equation (with not both zero) can be rewritten in any of the other named forms, by comparing coefficients:

6.4

Angle Between Two Straight Lines

Two straight lines in a plane are either parallel, coincident, or intersecting at exactly one point; when they intersect, they form two angles at that point — one acute (or a right angle) and its supp…

6.4.1

Condition for Parallel Lines

Lines in the same plane that never intersect are parallel. For and to be parallel, the angle between them must be (or ): .

6.4.2

Condition for Perpendicular Lines

For and to be perpendicular, the angle between them must be : taking the cotangent of gives , and forces , i.e. In general form, and are perpendicular exactly when .

6.4.3

Position of a Point with Respect to a Straight Line

Any line with splits the whole plane into two half-planes: the origin side (containing the origin) and the non-origin side.

6.4.4

Distance Formulas

Three distance formulas are developed:

6.4.5

Family of Lines

A family of lines is the set of all lines satisfying one common stated condition. Even though the equation of a straight line shows three symbols , dividing through by whichever of is nonzero shows th…

6.4.6

One Parameter Families

(i) arbitrary, fixed. Fixing the -intercept at, say, and letting range over the reals gives the family — every member shares the single point on the -axis, and different values of (e.g.

6.4.7

Two Parameters Families

(iii) Both arbitrary. The full two-parameter family (both constants free) is hard to visualise directly on a graph, but an important sub-case is easy: fixing a point and letting vary in sweeps out eve…

+Exercise 6.3i20 questions
  1. Q1Show that the lines $3x + 2y + 9 = 0$ and $12x + 8y - 15 = 0$ are parallel lines.Free
  2. Q2Find the equation of the straight line parallel to $5x - 4y + 3 = 0$ and having $x$-intercept $3$.Free
  3. Q3Find the distance between the line $4x + 3y + 4 = 0$ and the point (i) $(-2, 4)$ (ii) $(7, -3)$.Free
  4. Q4Write the equation of the lines through the point $(1, -1)$ (i) parallel to $x + 3y - 4 = 0$ (ii) perpendicular to $3x + 4y = 6$.Preview
  5. Q5If $(-4, 7)$ is one vertex of a rhombus and the equation of one diagonal is $5x - y + 7 = 0$, then find the equation of the other diagonal.Preview
  6. Q6Find the equation of the lines passing through the point of intersection of the lines $4x - y + 3 = 0$ and $5x + 2y + 7 = 0$, and (i) throug…Preview
  7. Q7Find the equations of two straight lines which are parallel to the line $12x + 5y + 2 = 0$ and at a unit distance from the point $(1, -1)$.Preview
  8. Q8Find the equations of straight lines which are perpendicular to the line $3x + 4y - 6 = 0$ and are at a distance of $4$ units from $(2, 1)$.Preview
  9. Q9Find the equation of a straight line parallel to $2x + 3y = 10$ and such that the sum of its intercepts on the axes is $15$.Preview
  10. Q10Find the length of the perpendicular and the co-ordinates of the foot of the perpendicular from $(-10, -2)$ to the line $x + y - 2 = 0$.Preview
  11. Q11If $p_1$ and $p_2$ are the lengths of the perpendiculars from the origin to the straight lines $x\sec\theta + y\csc\theta = 2a$ and $x\cos\t…Preview
  12. Q12Find the distance between the parallel lines (i) $12x + 5y = 7$ and $12x + 5y + 7 = 0$ (ii) $3x - 4y + 5 = 0$ and $6x - 8y - 15 = 0$.Preview
  13. Q13Find the family of straight lines (i) perpendicular to (ii) parallel to $3x + 4y - 12 = 0$.Preview
  14. Q14If the line joining the two points $A(2, 0)$ and $B(3, 1)$ is rotated about $A$ in the anticlockwise direction through an angle of $15^\circ…Preview
  15. Q15A ray of light coming from the point $(1, 2)$ is reflected at a point $A$ on the $x$-axis and it passes through the point $(5, 3)$. Find the…Preview
  16. Q16A line is drawn perpendicular to $5x = y + 7$. Find the equation of the line if the area of the triangle formed by this line with the co-ord…Preview
  17. Q17Find the image of the point $(-2, 3)$ about the line $x + 2y - 9 = 0$.Preview
  18. Q18A photocopy store charges ₹$1.50$ per copy for the first $10$ copies and ₹$1.00$ per copy after the $10$th copy. Let $x$ be the number of co…Preview
  19. Q19Find at least two equations of the straight lines in the family of the lines $y = 5x + b$, for which $b$ and the $x$-coordinate of the point…Preview
  20. Q20Find all the equations of the straight lines in the family of the lines $y = mx - 3$, for which $m$ and the $x$-coordinate of the point of i…Preview
6.5

Pair of Straight Lines

Two or more straight lines' equations can be packaged into a single equation of degree higher than one.

6.5.1

Pair of Lines Passing Through the Origin

The simplest case is a pair of lines both through the origin: and . Their combined equation is which suggests the general form: any homogeneous second-degree equation (every term of degree exactly ; '…

6.5.2

Angle Between a Pair of Straight Lines

For the pair of lines through the origin , dividing by and substituting gives , whose roots (the two slopes) satisfy, by the sum/product-of-roots relations, The angle between the two lines then follow…

6.5.3

Equation of the Bisectors of the Angle Between the Lines

The angle bisectors of the pair (i.e. of and , with ) are found as the locus of points equidistant from both lines.

6.5.4

General Form of a Pair of Straight Lines

43 Q

Multiplying out two arbitrary (not necessarily through the origin) lines and gives the general second-degree equation ast with — a non-homogeneous equation of degree two (unlike §6.5.1's homogeneous c…

+Exercise 6.4i18 questions
  1. Q1Find the combined equation of the straight lines whose separate equations are $x - 2y - 3 = 0$ and $x + y + 5 = 0$.Free
  2. Q2Show that $4x^2 + 4xy + y^2 - 6x - 3y - 4 = 0$ represents a pair of parallel lines.Free
  3. Q3Show that $2x^2 + 3xy - 2y^2 + 3x + y + 1 = 0$ represents a pair of perpendicular lines.Free
  4. Q4Show that the equation $2x^2 - xy - 3y^2 - 6x + 19y - 20 = 0$ represents a pair of intersecting lines. Show further that the angle between t…Preview
  5. Q5Prove that the equation to the straight lines through the origin, each of which makes an angle $\alpha$ with the straight line $y = x$, is $…Preview
  6. Q6Find the equation of the pair of straight lines passing through the point $(1, 3)$ and perpendicular to the lines $2x - 3y + 1 = 0$ and $5x…Preview
  7. Q7Find the separate equations of the following pair of straight lines (i) $3x^2 + 2xy - y^2 = 0$ (ii) $6(x-1)^2 + 5(x-1)(y-2) - 4(y-2)^2 = 0$…Preview
  8. Q8The slope of one of the straight lines $ax^2 + 2hxy + by^2 = 0$ is twice that of the other; show that $8h^2 = 9ab$.Preview
  9. Q9The slope of one of the straight lines $ax^2 + 2hxy + by^2 = 0$ is three times the other; show that $3h^2 = 4ab$.Preview
  10. Q10A $\triangle OPQ$ is formed by the pair of straight lines $x^2 - 4xy + y^2 = 0$ and the line $PQ$. The equation of $PQ$ is $x + y - 2 = 0$.…Preview
  11. Q11Find $p$ and $q$, if the following equation represents a pair of perpendicular lines: $6x^2 + 5xy - py^2 + 7x + qy - 5 = 0$.Preview
  12. Q12Find the value of $k$, if the following equation represents a pair of straight lines. Further, find whether these lines are parallel or inte…Preview
  13. Q13For what value of $k$ does the equation $12x^2 + 2kxy + 2y^2 + 11x - 5y + 2 = 0$ represent two straight lines?Preview
  14. Q14Show that the equation $9x^2 - 24xy + 16y^2 - 12x + 16y - 12 = 0$ represents a pair of parallel lines. Find the distance between them.Preview
  15. Q15Show that the equation $4x^2 + 4xy + y^2 - 6x - 3y - 4 = 0$ represents a pair of parallel lines. Find the distance between them.Preview
  16. Q16Prove that one of the straight lines given by $ax^2 + 2hxy + by^2 = 0$ will bisect the angle between the co-ordinate axes if $(a+b)^2 = 4h^2…Preview
  17. Q17If the pair of straight lines $x^2 - 2kxy - y^2 = 0$ bisects the angle between the pair of straight lines $x^2 - 2lxy - y^2 = 0$, show that…Preview
  18. Q18Prove that the straight lines joining the origin to the points of intersection of $3x^2 + 5xy - 3y^2 + 2x + 3y = 0$ and $3x - 2y - 1 = 0$ ar…Preview
+Exercise 6.5i25 questions
  1. Q1The equation of the locus of the point whose distance from the $y$-axis is half the distance from the origin is (1) $x^2+3y^2=0$ (2) $x^2-3y…Free
  2. Q2Which of the following equations is the locus of $(at^2, 2at)$? (1) $\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1$ (2) $\dfrac{x^2}{a^2}+\dfrac{y^2}{…Free
  3. Q3Which of the following points lies on the locus of $3x^2+3y^2-8x-12y+17=0$? (1) $(0,0)$ (2) $(-2,3)$ (3) $(1,2)$ (4) $(0,-1)$Free
  4. Q4If the point $(8,-5)$ lies on the locus $\dfrac{x^2}{16}-\dfrac{y^2}{25}=k$, then the value of $k$ is (1) $0$ (2) $1$ (3) $2$ (4) $3$Preview
  5. Q5The straight line joining the points $(2, 3)$ and $(-1, 4)$ passes through the point $(\alpha,\beta)$ if (1) $\alpha+2\beta=7$ (2) $3\alpha+…Preview
  6. Q6The slopes of the lines which make an angle $45^\circ$ with the line $3x-y=-5$ are (1) $1,-1$ (2) $\dfrac12,-2$ (3) $1,\dfrac12$ (4) $2,-\df…Preview
  7. Q7The equation of the straight line that forms an isosceles triangle with the coordinate axes in the first quadrant, with perimeter $4+2\sqrt2…Preview
  8. Q8The coordinates of the four vertices of a quadrilateral are $(-2,4)$, $(-1,2)$, $(1,2)$ and $(2,4)$, taken in order. The equation of the lin…Preview
  9. Q9The intercepts made on the coordinate axes by the perpendicular bisector of the line segment joining $(1, 2)$ and $(3,4)$ are (1) $5,-5$ (2)…Preview
  10. Q10The equation of the line with slope $2$ and length of the perpendicular from the origin equal to $\sqrt5$ is (1) $x+2y=\sqrt5$ (2) $2x+y=\sq…Preview
  11. Q11A line perpendicular to the line $5x-y=0$ forms a triangle with the coordinate axes. If the area of the triangle is $5$ sq. units, then its…Preview
  12. Q12The equation of the straight line perpendicular to the line $x-y+5=0$, through the point of intersection of the $y$-axis and the given line,…Preview
  13. Q13If the equation of the base, opposite to the vertex $(2, 3)$, of an equilateral triangle is $x+y=2$, then the length of a side is (1) $\sqrt…Preview
  14. Q14The line $(p+2q)x+(p-3q)y=p-q$, for different values of $p$ and $q$, passes through the point (1) $\left(\dfrac32,\dfrac52\right)$ (2) $\lef…Preview
  15. Q15The point on the line $2x-3y=5$ that is equidistant from $(1,2)$ and $(3,4)$ is (1) $(7,3)$ (2) $(4,1)$ (3) $(1,-1)$ (4) $(-2,3)$Preview
  16. Q16The image of the point $(2, 3)$ in the line $y=-x$ is (1) $(-3,-2)$ (2) $(-3,2)$ (3) $(-2,-3)$ (4) $(3,2)$Preview
  17. Q17The length of the perpendicular from the origin to the line $\dfrac{x}{3}-\dfrac{y}{4}=1$ is (1) $\dfrac{11}{5}$ (2) $\dfrac{5}{12}$ (3) $\d…Preview
  18. Q18The $y$-intercept of the straight line passing through $(1,3)$ and perpendicular to $2x-3y+1=0$ is (1) $\dfrac32$ (2) $\dfrac92$ (3) $\dfrac…Preview
  19. Q19If the two straight lines $x+(2k-7)y+3=0$ and $3kx+9y-5=0$ are perpendicular, then the value of $k$ is (1) $k=3$ (2) $k=\dfrac13$ (3) $k=\df…Preview
  20. Q20If a vertex of a square is at the origin and one of its sides lies along the line $4x+3y-20=0$, then the area of the square is (1) $20$ sq.…Preview
  21. Q21If the lines represented by the equation $6x^2+41xy-7y^2=0$ make angles $\alpha$ and $\beta$ with the $x$-axis, then $\tan\alpha\tan\beta=$…Preview
  22. Q22The area of the triangle formed by the lines $x^2-4y^2=0$ and $x=a$ is (1) $2a^2$ (2) $\dfrac{\sqrt3}{2}a^2$ (3) $\dfrac12a^2$ (4) $2\sqrt3\…Preview
  23. Q23If one of the lines given by $6x^2-xy+4cy^2=0$ is $3x+4y=0$, then $c$ equals (1) $-3$ (2) $-1$ (3) $3$ (4) $1$Preview
  24. Q24$\theta$ is the acute angle between the lines $x^2-xy-6y^2=0$. Then $\dfrac{2\cos\theta+3\sin\theta}{4\sin\theta+5\cos\theta}$ is (1) $1$ (2…Preview
  25. Q25The equation of one of the lines represented by the equation $x^2+2xy\cot\theta-y^2=0$ is (1) $x-y\cot\theta=0$ (2) $x+y\tan\theta=0$ (3) $x…Preview

Sample & Board Papers

Sample papers and previous-year board questions for this subject.

+Show 36 questions36 questions
  1. Q1Which of the following has the greatest y-intercept in magnitude? (a) $3x + 4y = 5$ (b) $2x + 3y = 4$ (c) $4x + 5y = 6$ (d) $x + 2y = 3$Preview
  2. Q2If the circle has both x and y axes as tangents and has radius 1 unit then the equation of the circle is: (a) $(x+1)^2 + (y+1)^2 = 1$ (b) $x…Preview
  3. Q3Find the point on x-axis which is equidistant from the points (7, -6) and (3, 4).Preview
  4. Q4Find the equation of the straight line which passes through the intersection of the straight lines $5x-6y=1$ and $3x+2y+5=0$ and is perpendi…Preview
  5. Q5Show that $x^2 - y^2 + x - 3y - 2 = 0$ represents a pair of straight lines. Also find the angle between them.Preview
  6. Q6(a) Show that the circles $x^2+y^2-2x+6y+6=0$ and $x^2+y^2-5x+6y+15=0$ touch each other. **OR** (b) Evaluate: $\displaystyle\lim_{x\to 0} \d…Preview
  7. Q7The line $\dfrac{x}{a}-\dfrac{y}{b}=0$ has the slope 1, if: (a) $a=b$ (b) only for $a=1, b=1$ (c) $a>b$ (d) $a<b$Preview
  8. Q8The straight line joining the points $(2, 3)$ and $(-1, 4)$ passes through $(\alpha, \beta)$ if: (a) $\alpha+3\beta=11$ (b) $3\alpha+\beta=1…Preview
  9. Q9Find the separate equations of straight lines from a combined equation $2x^2+xy-3y^2=0$.Preview
  10. Q10Find the nearest point on the line $x-2y=5$ from the origin.Preview
  11. Q11If the point $(8, -5)$ lies on the locus $\dfrac{x^2}{16} - \dfrac{y^2}{25} = k$, then the value of $k$ is: (a) 0 (b) 1 (c) 2 (d) 3Preview
  12. Q12Equation of the straight line perpendicular to the line $x - y + 5 = 0$, through the point of intersection on the $y$ axis and the given lin…Preview
  13. Q13The length of the perpendicular drawn from the origin to a line is 12 and makes an angle $150°$ with positive direction of the $x$-axis. Fin…Preview
  14. Q14If the line joining two points $A(2, 0)$ and $B(3, 1)$ is rotated about A in anticlockwise direction through an angle of $15°$, then find th…Preview
  15. Q15The slope of the line which makes an angle $45°$ with the line $3x-y=-5$ are: (a) $1, \frac{1}{2}$ (b) $1, -1$ (c) $2, \frac{-1}{2}$ (d) $\f…Preview
  16. Q16Find the equation of the line passing through the points $(1, 1)$ and $(-2, 3)$.Preview
  17. Q17Find the distance from the point $(1, 2)$ to the line $5x+12y-3=0$.Preview
  18. Q18(a) For what value of $k$ does the equation $12x^2+2kxy+2y^2+11x-5y+2=0$ represent two straight lines. **OR** (b) Evaluate: $\displaystyle\l…Preview
  19. Q19The points lie on the locus of $3x^2 + 3y^2 - 8x - 12y + 17 = 0$. (a) (1, 2) (b) (0, 0) (c) (0, -1) (d) (-2, 3)Preview
  20. Q20If one of the lines given by $6x^2 - xy + 4cy^2 = 0$ is $3x + 4y = 0$, then $c$ equals to: (a) 3 (b) $-3$ (c) 1 (d) $-1$Preview
  21. Q21Find the separate equation of the pair of straight lines $3x^2 + 2xy - y^2 = 0$.Preview
  22. Q22If $\theta$ is a parameter, find the equation of the locus of a moving point, whose coordinates are $x = a\cos^3\theta$, $y = a\sin^3\theta$…Preview
  23. Q23If the point $(8,-5)$ lies on the locus $\dfrac{x^2}{16}-\dfrac{y^2}{25}=k$, then the value of $k$ is: (a) $2$ (b) $0$ (c) $3$ (d) $1$Preview
  24. Q24The area of the triangle formed by the lines $x^2-4y^2=0$ and $x=a$ is: (a) $\dfrac{1}{2}a^2$ (b) $2a^2$ (c) $\dfrac{2}{\sqrt3}a^2$ (d) $\df…Preview
  25. Q25If the points $(x,-2)$, $(5,2)$, $(8,8)$ are collinear, then $x$ is equal to: (a) $1$ (b) $-3$ (c) $3$ (d) $\dfrac{1}{3}$Preview
  26. Q26Show that the lines are $3x+2y+9=0$ and $12x+8y-15=0$ are parallel lines.Preview
  27. Q27Find the separate equation of the pair of straight lines $5x^2+6xy+y^2=0$.Preview
  28. Q28If the points $(x, -2), (5, 2), (8, 8)$ are collinear, then $x$ is equal to: (a) $1$ (b) $-3$ (c) $3$ (d) $\dfrac{1}{3}$Preview
  29. Q29Straight line joining the points $(2, 3)$ and $(-1, 4)$ passes through the point $(\alpha, \beta)$ if: (a) $\alpha + 3\beta = 11$ (b) $\alph…Preview
  30. Q30The equation of the line through the point $(1, -1)$ and perpendicular to $3x + 4y = 6$ is: (a) $4x+3y+7=0$ (b) $4x-3y-7=0$ (c) $3x+4y+7=0$…Preview
  31. Q31Find the equation of the locus of P, if for all values of $\alpha$, the co-ordinates of a moving point P is $(9\cos\alpha, 9\sin\alpha)$, wh…Preview
  32. Q32Show that the points $\left(0, -\dfrac{3}{2}\right)$, $(1, -1)$ and $\left(2, -\dfrac{1}{2}\right)$ are collinear.Preview
  33. Q33Which of the following equations is the locus of $(a\cos\theta, b\sin\theta)$? (a) $x^2+y^2=a^2$ (b) $\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1$ (…Preview
  34. Q34The image of the point $(2, 3)$ in the line $y = -x$ is: (a) $(-2, -3)$ (b) $(-3, -2)$ (c) $(3, 2)$ (d) $(-3, 2)$Preview
  35. Q35Find the distance from a point $(1, 2)$ to the line $5x + 12y - 3 = 0$.Preview
  36. Q36Rewrite $\sqrt{3}x + y + 4 = 0$ into normal form.Preview