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Mathematics · Class 11 Science

Ch 9Straight Lines — Class 11 Mathematics, concept-first.

Coordinate geometry is the bridge between algebra and geometry. You have already encountered it in earlier classes — plotting points, finding distances, dividing line segments, and computing areas.

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Coordinate Geometry

Imagine you're telling a friend where you left your book in a library. You don't say "near the window" — you say "third shelf, second row, fourth book from the left." You're using numbers to pin down an exact location.

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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.

9.1

Introduction

Coordinate geometry is the bridge between algebra and geometry. You have already encountered it in earlier classes — plotting points, finding distances, dividing line segments, and computing areas.

9.2

Slope of a Line

When a line is drawn in a coordinate plane, it makes two angles with the x‑axis — one acute and one obtuse, which are supplementary (they add up to ).

9.2.1

Slope of a Line When Coordinates of Any Two Points on the Line Are Given

A line is completely determined by any two points on it. Given two points, we can find the line's slope — the measure of its steepness — without needing a diagram.

9.2.2

Conditions for Parallelism and Perpendicularity of Lines in Terms of Their Slopes

When two lines lie in the same coordinate plane, their slopes tell us everything about whether they are parallel or perpendicular — provided neither line is vertical.

+Worked Examplesi1 question
  1. Example 1Find the slope of the lines: (a) Passing through the points $(3, -2)$ and $(-1, 4)$, (b) Passing through the points $(3, -2)$ and $(7, -2)$,…Preview
9.2.3

Angle Between Two Lines

13 Q

When we work with just one line in a plane, we talk about its slope and inclination. But the moment we consider two lines, a natural question arises: what is the angle between them? Two lines in a pla…

+Worked Examplesi2 questions
  1. Example 2If the angle between two lines is $\dfrac{\pi}{4}$ and slope of one of the lines is $\dfrac{1}{2}$, find the slope of the other line.Free
  2. Example 3Line through the points $(-2, 6)$ and $(4, 8)$ is perpendicular to the line through the points $(8, 12)$ and $(x, 24)$. Find the value of $x…Preview
+Exercise 9.1i11 questions
  1. Q1Draw a quadrilateral in the Cartesian plane, whose vertices are $(-4, 5)$, $(0, 7)$, $(5, -5)$ and $(-4, -2)$. Also, find its area.Free
  2. Q2The base of an equilateral triangle with side $2a$ lies along the y-axis such that the mid-point of the base is at the origin. Find vertices…Free
  3. Q3Find the distance between $P(x_1, y_1)$ and $Q(x_2, y_2)$ when: (i) $PQ$ is parallel to the y-axis, (ii) $PQ$ is parallel to the x-axis.Free
  4. Q4Find a point on the x-axis, which is equidistant from the points $(7, 6)$ and $(3, 4)$.Preview
  5. Q5Find the slope of a line, which passes through the origin, and the mid-point of the line segment joining the points $P(0, -4)$ and $B(8, 0)$…Preview
  6. Q6Without using the Pythagoras theorem, show that the points $(4, 4)$, $(3, 5)$ and $(-1, -1)$ are the vertices of a right angled triangle.Preview
  7. Q7Find the slope of the line, which makes an angle of $30^\circ$ with the positive direction of y-axis measured anticlockwise.Preview
  8. Q8Without using distance formula, show that points $(-2, -1)$, $(4, 0)$, $(3, 3)$ and $(-3, 2)$ are the vertices of a parallelogram.Preview
  9. Q9Find the angle between the x-axis and the line joining the points $(3, -1)$ and $(4, -2)$.Preview
  10. Q10The slope of a line is double of the slope of another line. If tangent of the angle between them is $\dfrac{1}{3}$, find the slopes of the l…Preview
  11. Q11A line passes through $(x_1, y_1)$ and $(h, k)$. If slope of the line is $m$, show that $k - y_1 = m(h - x_1)$.Preview
9.3

Various Forms of the Equation of a Line

Every line in a plane contains infinitely many points. The central question is: given a line and an arbitrary point in the -plane, how do we decide whether lies on ? The answer is that we need a condi…

9.3.1

Horizontal and Vertical Lines

When a line is horizontal, every point on it has the same y-coordinate. If a horizontal line L is at a distance from the x-axis, then the y-coordinate of every point on L is either or .

+Worked Examplesi1 question
  1. Example 4Find the equations of the lines parallel to axes and passing through $(-2, 3)$.Preview
9.3.2

Point-slope Form

We now move from the slope-intercept form, which requires the y-intercept, to a more general method. Suppose you know a line's slope and any one point that lies on it — not necessarily where it crosse…

+Worked Examplesi1 question
  1. Example 5Find the equation of the line through $(-2, 3)$ with slope $-4$.Preview
9.3.3

Two-point Form

When you know two distinct points that lie on a line, you can write its equation directly. This is one of the most practical forms because you don't need to calculate the slope separately — the formul…

+Worked Examplesi1 question
  1. Example 6Write the equation of the line through the points $(1, -1)$ and $(3, 5)$.Preview
9.3.4

Slope-intercept Form

When a line's slope is known and we also know where it cuts one of the coordinate axes, we can write its equation directly.

+Worked Examplesi1 question
  1. Example 7Write the equation of the lines for which $\tan\theta = \dfrac{1}{2}$, where $\theta$ is the inclination of the line and (i) y-intercept is…Preview
9.3.5

Intercept - Form

20 Q

When a line crosses the coordinate axes, the points where it meets the x-axis and y-axis are especially easy to work with.

+Worked Examplesi1 question
  1. Example 8Find the equation of the line, which makes intercepts $-3$ and $2$ on the x- and y-axes respectively.Preview
+Exercise 9.2i19 questions
  1. Q1Write the equations for the x- and y-axes.Free
  2. Q2Passing through the point $(-4, 3)$ with slope $\dfrac{1}{2}$.Free
  3. Q3Passing through $(0, 0)$ with slope $m$.Free
  4. Q4Passing through $(2, 2\sqrt{3})$ and inclined with the x-axis at an angle of $75^\circ$.Preview
  5. Q5Intersecting the x-axis at a distance of $3$ units to the left of origin with slope $-2$.Preview
  6. Q6Intersecting the y-axis at a distance of $2$ units above the origin and making an angle of $30^\circ$ with positive direction of the x-axis.Preview
  7. Q7Passing through the points $(-1, 1)$ and $(2, -4)$.Preview
  8. Q8The vertices of $\triangle PQR$ are $P(2, 1)$, $Q(-2, 3)$ and $R(4, 5)$. Find equation of the median through the vertex $R$.Preview
  9. Q9Find the equation of the line passing through $(-3, 5)$ and perpendicular to the line through the points $(2, 5)$ and $(-3, 6)$.Preview
  10. Q10A line perpendicular to the line segment joining the points $(1, 0)$ and $(2, 3)$ divides it in the ratio $1 : n$. Find the equation of the…Preview
  11. Q11Find the equation of a line that cuts off equal intercepts on the coordinate axes and passes through the point $(2, 3)$.Preview
  12. Q12Find equation of the line passing through the point $(2, 2)$ and cutting off intercepts on the axes whose sum is $9$.Preview
  13. Q13Find equation of the line through the point $(0, 2)$ making an angle $\dfrac{2\pi}{3}$ with the positive x-axis. Also, find the equation of…Preview
  14. Q14The perpendicular from the origin to a line meets it at the point $(-2, 9)$, find the equation of the line.Preview
  15. Q15The length $L$ (in centimetre) of a copper rod is a linear function of its Celsius temperature $C$. In an experiment, if $L = 124.942$ when…Preview
  16. Q16The owner of a milk store finds that, he can sell $980$ litres of milk each week at Rs $14$/litre and $1220$ litres of milk each week at Rs…Preview
  17. Q17$P(a, b)$ is the mid-point of a line segment between axes. Show that equation of the line is $\dfrac{x}{a} + \dfrac{y}{b} = 2$.Preview
  18. Q18Point $R(h, k)$ divides a line segment between the axes in the ratio $1 : 2$. Find equation of the line.Preview
  19. Q19By using the concept of equation of a line, prove that the three points $(3, 0)$, $(-2, -2)$ and $(8, 2)$ are collinear.Preview
9.4

Distance of a Point From a Line

The distance of a point from a line is defined as the length of the perpendicular drawn from the point to the line. This is the shortest possible distance between the point and any point on the line.

9.4.1

Distance Between Two Parallel Lines

19 Q

The idea is simple: two parallel lines never meet, so the distance between them is constant — the length of the perpendicular segment joining them.

+Worked Examplesi2 questions
  1. Example 9Find the distance of the point $(3, -5)$ from the line $3x - 4y - 26 = 0$.Free
  2. Example 10Find the distance between the parallel lines $3x - 4y + 7 = 0$ and $3x - 4y + 5 = 0$.Preview
+Exercise 9.3i17 questions
  1. Q1Reduce the following equations into slope-intercept form and find their slopes and the y-intercepts. (i) $x + 7y = 0$, (ii) $6x + 3y - 5 = 0…Free
  2. Q2Reduce the following equations into intercept form and find their intercepts on the axes. (i) $3x + 2y - 12 = 0$, (ii) $4x - 3y = 6$, (iii)…Free
  3. Q3Find the distance of the point $(-1, 1)$ from the line $12(x + 6) = 5(y - 2)$.Free
  4. Q4Find the points on the x-axis, whose distances from the line $\dfrac{x}{3} + \dfrac{y}{4} = 1$ are $4$ units.Preview
  5. Q5Find the distance between parallel lines (i) $15x + 8y - 34 = 0$ and $15x + 8y + 31 = 0$ (ii) $l(x + y) + p = 0$ and $l(x + y) - r = 0$.Preview
  6. Q6Find equation of the line parallel to the line $3x - 4y + 2 = 0$ and passing through the point $(-2, 3)$.Preview
  7. Q7Find equation of the line perpendicular to the line $x - 7y + 5 = 0$ and having x-intercept $3$.Preview
  8. Q8Find angles between the lines $\sqrt{3}\,x + y = 1$ and $x + \sqrt{3}\,y = 1$.Preview
  9. Q9The line through the points $(h, 3)$ and $(4, 1)$ intersects the line $7x - 9y - 19 = 0$ at right angle. Find the value of $h$.Preview
  10. Q10Prove that the line through the point $(x_1, y_1)$ and parallel to the line $Ax + By + C = 0$ is $A(x - x_1) + B(y - y_1) = 0$.Preview
  11. Q11Two lines passing through the point $(2, 3)$ intersects each other at an angle of $60^\circ$. If slope of one line is $2$, find equation of…Preview
  12. Q12Find the equation of the right bisector of the line segment joining the points $(3, 4)$ and $(-1, 2)$.Preview
  13. Q13Find the coordinates of the foot of perpendicular from the point $(-1, 3)$ to the line $3x - 4y - 16 = 0$.Preview
  14. Q14The perpendicular from the origin to the line $y = mx + c$ meets it at the point $(-1, 2)$. Find the values of $m$ and $c$.Preview
  15. Q15If $p$ and $q$ are the lengths of perpendiculars from the origin to the lines $x\cos\theta - y\sin\theta = k\cos 2\theta$ and $x\sec\theta +…Preview
  16. Q16In the triangle $ABC$ with vertices $A(2, 3)$, $B(4, -1)$ and $C(1, 2)$, find the equation and length of altitude from the vertex $A$.Preview
  17. Q17If $p$ is the length of perpendicular from the origin to the line whose intercepts on the axes are $a$ and $b$, then show that $\dfrac{1}{p^…Preview

Miscellaneous Examples

+Miscellaneous Examplesi6 questions
  1. Example 11If the lines $2x + y - 3 = 0$, $5x + ky - 3 = 0$ and $3x - y - 2 = 0$ are concurrent, find the value of $k$.Free
  2. Example 12Find the distance of the line $4x - y = 0$ from the point $P(4, 1)$ measured along the line making an angle of $135^\circ$ with the positive…Free
  3. Example 13Assuming that straight lines work as the plane mirror for a point, find the image of the point $(1, 2)$ in the line $x - 3y + 4 = 0$.Preview
  4. Example 14Show that the area of the triangle formed by the lines $y = m_1 x + c_1$, $y = m_2 x + c_2$ and $x = 0$ is $\dfrac{(c_1 - c_2)^2}{2\,|m_1 -…Preview
  5. Example 15A line is such that its segment between the lines $5x - y + 4 = 0$ and $3x + 4y - 4 = 0$ is bisected at the point $(1, 5)$. Obtain its equat…Preview
  6. Example 16Show that the path of a moving point such that its distances from two lines $3x - 2y = 5$ and $3x + 2y = 5$ are equal is a straight line.Preview

Miscellaneous Exercise on Chapter 9

+Miscellaneous Exercisei23 questions
  1. Q1Find the values of $k$ for which the line $(k - 3)x - (4 - k^2)y + k^2 - 7k + 6 = 0$ is (a) Parallel to the x-axis, (b) Parallel to the y-ax…Free
  2. Q2Find the equations of the lines, which cut-off intercepts on the axes whose sum and product are $1$ and $-6$, respectively.Free
  3. Q3What are the points on the y-axis whose distance from the line $\dfrac{x}{3} + \dfrac{y}{4} = 1$ is $4$ units.Free
  4. Q4Find perpendicular distance from the origin to the line joining the points $(\cos\theta, \sin\theta)$ and $(\cos\phi, \sin\phi)$.Preview
  5. Q5Find the equation of the line parallel to y-axis and drawn through the point of intersection of the lines $x - 7y + 5 = 0$ and $3x + y = 0$.Preview
  6. Q6Find the equation of a line drawn perpendicular to the line $\dfrac{x}{4} + \dfrac{y}{6} = 1$ through the point, where it meets the y-axis.Preview
  7. Q7Find the area of the triangle formed by the lines $y - x = 0$, $x + y = 0$ and $x - k = 0$.Preview
  8. Q8Find the value of $p$ so that the three lines $3x + y - 2 = 0$, $px + 2y - 3 = 0$ and $2x - y - 3 = 0$ may intersect at one point.Preview
  9. Q9If three lines whose equations are $y = m_1 x + c_1$, $y = m_2 x + c_2$ and $y = m_3 x + c_3$ are concurrent, then show that $m_1(c_2 - c_3)…Preview
  10. Q10Find the equation of the lines through the point $(3, 2)$ which make an angle of $45^\circ$ with the line $x - 2y = 3$.Preview
  11. Q11Find the equation of the line passing through the point of intersection of the lines $4x + 7y - 3 = 0$ and $2x - 3y + 1 = 0$ that has equal…Preview
  12. Q12Show that the equation of the line passing through the origin and making an angle $\theta$ with the line $y = mx + c$ is $\dfrac{y}{x} = \df…Preview
  13. Q13In what ratio, the line joining $(-1, 1)$ and $(5, 7)$ is divided by the line $x + y = 4$?Preview
  14. Q14Find the distance of the line $4x + 7y + 5 = 0$ from the point $(1, 2)$ along the line $2x - y = 0$.Preview
  15. Q15Find the direction in which a straight line must be drawn through the point $(-1, 2)$ so that its point of intersection with the line $x + y…Preview
  16. Q16The hypotenuse of a right angled triangle has its ends at the points $(1, 3)$ and $(-4, 1)$. Find an equation of the legs (perpendicular sid…Preview
  17. Q17Find the image of the point $(3, 8)$ with respect to the line $x + 3y = 7$ assuming the line to be a plane mirror.Preview
  18. Q18If the lines $y = 3x + 1$ and $2y = x + 3$ are equally inclined to the line $y = mx + 4$, find the value of $m$.Preview
  19. Q19If sum of the perpendicular distances of a variable point $P(x, y)$ from the lines $x + y - 5 = 0$ and $3x - 2y + 7 = 0$ is always $10$. Sho…Preview
  20. Q20Find equation of the line which is equidistant from parallel lines $9x + 6y - 7 = 0$ and $3x + 2y + 6 = 0$.Preview
  21. Q21A ray of light passing through the point $(1, 2)$ reflects on the x-axis at point $A$ and the reflected ray passes through the point $(5, 3)…Preview
  22. Q22Prove that the product of the lengths of the perpendiculars drawn from the points $\left(\sqrt{a^2 - b^2},\, 0\right)$ and $\left(-\sqrt{a^2…Preview
  23. Q23A person standing at the junction (crossing) of two straight paths represented by the equations $2x - 3y + 4 = 0$ and $3x + 4y - 5 = 0$ want…Preview

Summary

- Slope of a line: where is the inclination (). For two points and , . - Parallel lines: . Perpendicular lines: . - Angle between two lines: .

NCERT Exemplar

Higher-order thinking problems from the NCERT Exemplar.

+Show 59 questions59 questions
  1. Q1Find the equation of the straight line which passes through the point $(1,-2)$ and cuts off equal intercepts from axes.Free
  2. Q2Find the equation of the line passing through the point $(5,2)$ and perpendicular to the line joining the points $(2,3)$ and $(3,-1)$.Free
  3. Q3Find the angle between the lines $y=(2-\sqrt{3})(x+5)$ and $y=(2+\sqrt{3})(x-7)$.Free
  4. Q4Find the equation of the lines which passes through the point $(3,4)$ and cuts off intercepts from the coordinate axes such that their sum i…Preview
  5. Q5Find the points on the line $x+y=4$ which lie at a unit distance from the line $4x+3y=10$.Preview
  6. Q6Show that the tangent of an angle between the lines $\dfrac{x}{a}+\dfrac{y}{b}=1$ and $\dfrac{x}{a}-\dfrac{y}{b}=1$ is $\dfrac{2ab}{a^2-b^2}…Preview
  7. Q7Find the equation of lines passing through $(1,2)$ and making angle $30^\circ$ with $y$-axis.Preview
  8. Q8Find the equation of the line passing through the point of intersection of $2x+y=5$ and $x+3y+8=0$ and parallel to the line $3x+4y=7$.Preview
  9. Q9For what values of $a$ and $b$ the intercepts cut off on the coordinate axes by the line $ax+by+8=0$ are equal in length but opposite in sig…Preview
  10. Q10If the intercept of a line between the coordinate axes is divided by the point $(-5,4)$ in the ratio $1:2$, then find the equation of the li…Preview
  11. Q11Find the equation of a straight line on which length of perpendicular from the origin is four units and the line makes an angle of $120^\cir…Preview
  12. Q12Find the equation of one of the sides of an isosceles right angled triangle whose hypotenuse is given by $3x+4y=4$ and the opposite vertex o…Preview
  13. Q13If the equation of the base of an equilateral triangle is $x+y=2$ and the vertex is $(2,-1)$, then find the length of the side of the triang…Preview
  14. Q14A variable line passes through a fixed point P. The algebraic sum of the perpendiculars drawn from the points $(2,0)$, $(0,2)$ and $(1,1)$ o…Preview
  15. Q15In what direction should a line be drawn through the point $(1,2)$ so that its point of intersection with the line $x+y=4$ is at a distance…Preview
  16. Q16A straight line moves so that the sum of the reciprocals of its intercepts made on axes is constant. Show that the line passes through a fix…Preview
  17. Q17Find the equation of the line which passes through the point $(-4,3)$ and the portion of the line intercepted between the axes is divided in…Preview
  18. Q18Find the equations of the lines through the point of intersection of the lines $x-y+1=0$ and $2x-3y+5=0$ and whose distance from the point $…Preview
  19. Q19If the sum of the distances of a moving point in a plane from the axes is 1, then find the locus of the point.Preview
  20. Q20$P_1$, $P_2$ are points on either of the two lines $y-\sqrt{3}\,|x|=2$ at a distance of 5 units from their point of intersection. Find the c…Preview
  21. Q21If $p$ is the length of perpendicular from the origin on the line $\dfrac{x}{a}+\dfrac{y}{b}=1$ and $a^2$, $p^2$, $b^2$ are in A.P, then sho…Preview
  22. Q22A line cutting off intercept $-3$ from the $y$-axis and the tangent at angle to the $x$-axis is $\dfrac{3}{5}$, its equation is (A) $5y-3x+1…Preview
  23. Q23Slope of a line which cuts off intercepts of equal lengths on the axes is (A) $-1$ (B) $-0$ (C) $2$ (D) $\sqrt{3}$Preview
  24. Q24The equation of the straight line passing through the point $(3,2)$ and perpendicular to the line $y=x$ is (A) $x-y=5$ (B) $x+y=5$ (C) $x+y=…Preview
  25. Q25The equation of the line passing through the point $(1,2)$ and perpendicular to the line $x+y+1=0$ is (A) $y-x+1=0$ (B) $y-x-1=0$ (C) $y-x+2…Preview
  26. Q26The tangent of angle between the lines whose intercepts on the axes are $a,-b$ and $b,-a$, respectively, is (A) $\dfrac{a^2-b^2}{ab}$ (B) $\…Preview
  27. Q27If the line $\dfrac{x}{a}+\dfrac{y}{b}=1$ passes through the points $(2,-3)$ and $(4,-5)$, then $(a,b)$ is (A) $(1,1)$ (B) $(-1,1)$ (C) $(1,…Preview
  28. Q28The distance of the point of intersection of the lines $2x-3y+5=0$ and $3x+4y=0$ from the line $5x-2y=0$ is (A) $\dfrac{130}{17\sqrt{29}}$ (…Preview
  29. Q29The equations of the lines which pass through the point $(3,-2)$ and are inclined at $60^\circ$ to the line $\sqrt{3}x+y=1$ is (A) $y+2=0,\…Preview
  30. Q30The equations of the lines passing through the point $(1,0)$ and at a distance $\dfrac{\sqrt{3}}{2}$ from the origin, are (A) $\sqrt{3}x+y-\…Preview
  31. Q31The distance between the lines $y=mx+c_1$ and $y=mx+c_2$ is (A) $\dfrac{c_1-c_2}{\sqrt{m^2+1}}$ (B) $\dfrac{|c_1-c_2|}{\sqrt{1+m^2}}$ (C) $\…Preview
  32. Q32The coordinates of the foot of perpendiculars from the point $(2,3)$ on the line $y=3x+4$ is given by (A) $\left(\dfrac{37}{10},\dfrac{-1}{1…Preview
  33. Q33If the coordinates of the middle point of the portion of a line intercepted between the coordinate axes is $(3,2)$, then the equation of the…Preview
  34. Q34Equation of the line passing through $(1,2)$ and parallel to the line $y=3x-1$ is (A) $y+2=x+1$ (B) $y+2=3(x+1)$ (C) $y-2=3(x-1)$ (D) $y-2=x…Preview
  35. Q35Equations of diagonals of the square formed by the lines $x=0$, $y=0$, $x=1$ and $y=1$ are (A) $y=x,\ y+x=1$ (B) $y=x,\ x+y=2$ (C) $2y=x,\ y…Preview
  36. Q36For specifying a straight line, how many geometrical parameters should be known? (A) $1$ (B) $2$ (C) $4$ (D) $3$Preview
  37. Q37The point $(4,1)$ undergoes the following two successive transformations: (i) Reflection about the line $y=x$ (ii) Translation through a dis…Preview
  38. Q38A point equidistant from the lines $4x+3y+10=0$, $5x-12y+26=0$ and $7x+24y-50=0$ is (A) $(1,-1)$ (B) $(1,1)$ (C) $(0,0)$ (D) $(0,1)$Preview
  39. Q39A line passes through $(2,2)$ and is perpendicular to the line $3x+y=3$. Its $y$-intercept is (A) $\dfrac{1}{3}$ (B) $\dfrac{2}{3}$ (C) $1$…Preview
  40. Q40The ratio in which the line $3x+4y+2=0$ divides the distance between the lines $3x+4y+5=0$ and $3x+4y-5=0$ is (A) $1:2$ (B) $3:7$ (C) $2:3$…Preview
  41. Q41One vertex of the equilateral triangle with centroid at the origin and one side as $x+y-2=0$ is (A) $(-1,-1)$ (B) $(2,2)$ (C) $(-2,-2)$ (D)…Preview
  42. Q42If $a,b,c$ are in A.P., then the straight lines $ax+by+c=0$ will always pass through ____.Preview
  43. Q43The line which cuts off equal intercept from the axes and pass through the point $(1,-2)$ is ____.Preview
  44. Q44Equations of the lines through the point $(3,2)$ and making an angle of $45^\circ$ with the line $x-2y=3$ are ____.Preview
  45. Q45The points $(3,4)$ and $(2,-6)$ are situated on the ____ of the line $3x-4y-8=0$.Preview
  46. Q46A point moves so that square of its distance from the point $(3,-2)$ is numerically equal to its distance from the line $5x-12y=3$. The equa…Preview
  47. Q47Locus of the mid-points of the portion of the line $x\sin\theta + y\cos\theta = p$ intercepted between the axes is ____.Preview
  48. Q48If the vertices of a triangle have integral coordinates, then the triangle can not be equilateral.Preview
  49. Q49The points $A(-2,1)$, $B(0,5)$, $C(-1,2)$ are collinear.Preview
  50. Q50Equation of the line passing through the point $(a\cos^3\theta,\ a\sin^3\theta)$ and perpendicular to the line $x\sec\theta + y\,\mathrm{cos…Preview
  51. Q51The straight line $5x+4y=0$ passes through the point of intersection of the straight lines $x+2y-10=0$ and $2x+y+5=0$.Preview
  52. Q52The vertex of an equilateral triangle is $(2,3)$ and the equation of the opposite side is $x+y=2$. Then the other two sides are $y-3=(2\pm\s…Preview
  53. Q53The equation of the line joining the point $(3,5)$ to the point of intersection of the lines $4x+y-1=0$ and $7x-3y-35=0$ is equidistant from…Preview
  54. Q54The line $\dfrac{x}{a}+\dfrac{y}{b}=1$ moves in such a way that $\dfrac{1}{a^2}+\dfrac{1}{b^2}=\dfrac{1}{c^2}$, where $c$ is a constant. The…Preview
  55. Q55The lines $ax+2y+1=0$, $bx+3y+1=0$ and $cx+4y+1=0$ are concurrent if $a,b,c$ are in G.P.Preview
  56. Q56Line joining the points $(3,-4)$ and $(-2,6)$ is perpendicular to the line joining the points $(-3,6)$ and $(9,-18)$.Preview
  57. Q57Match the entries of Column $C_1$ with their appropriate answers given under Column $C_2$. Column $C_1$: (a) The coordinates of the points P…Preview
  58. Q58The value of $\lambda$, if the lines $(2x+3y+4)+\lambda(6x-y+12)=0$ are — match Column $C_1$ with Column $C_2$. Column $C_1$: (a) parallel t…Preview
  59. Q59The equation of the line through the intersection of the lines $2x-3y=0$ and $4x-5y=2$ and — match Column $C_1$ with Column $C_2$. Column $C…Preview