1. The document contains 35 problems related to straight lines and conic sections like circles, ellipses, parabolas and hyperbolas. The problems involve finding equations of lines and conic sections given certain properties, finding properties like foci, vertices, axes, etc. from given equations. 2. Specific problems include finding slopes of lines, angle between lines, perpendicular lines, finding equations of lines passing through points or perpendicular/parallel to other lines. 3. For conic sections, problems include finding equations given foci, directrix, passing points, intersecting lines, centers and radii of circles, lengths of axes, foci, latus rectum and eccentricities of ellipses and hyperbol

1. CHAPTER – 10
THE STRAIGHT LINES
1. If A (-2, 1), B (2, 3) and C (-2, -4) are three points, find the angle between BA and BC.
2. Find the angle between the line joining the points (0, 0), (2, 3) and the points (2, -2), (3, 5).
3. Let A (6, 4) and B (2, 12) be two given points. Find the slope of a line perpendicular to AB.
4. Determine x so that 2 is the slope of the line through (2, 5) and (x, 3).
5. If the angle between two lines is
𝜋
4
and slope of one of the line
1
2
, find the slope of the other line.
6. If points (a, 0), (0, b) and P(x, y) be the given collinear, using the concept of slope prove that
𝑥
𝑎
+
𝑦
𝑏
= 1.
7. Find the slopes of the lines which make the following angles with the positive direction of x – axis :
(i) -
𝜋
4
(ii)
2𝜋
3
(iii)
3𝜋
4
(iv)
𝜋
3
8. The slope of a line is double of the slope of another line. If tangents of the angle between them is
1
3
, find the
sopes of the other line.
9. Line 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.
10. Find the equation of a line that y – intercept 4 and is perpendicular to the line joining (2, -3) and (4,2)
11. Find the equation of a line passing through (2, -3) and inclined at an angle of 135o
with the positive direction of x
– axis.
12. The perpendicular from the origin to a line meets it at the point (-2, 9), find the equation of the line.
13. Two lines passing through the point (2, 3) intersect each other at an angle of 60o
. If slope of one line is 2, find the
equation of the other line.
14. Find the equation of the perpendicular bisector of the line segment joining the points (1, 1) and (2,3).
15. A 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 line.
16. The vertices of a triangle are A(10, 4), B (-4, 9) and C (-2, -1). Find the equation of its altitudes. Also, find its
orthocentre.
17. Find the equation of the right bisector of the line segment joining the points (3, 4) and (-1, 2).
18. In what ratio, the line joining (-1, 1) and (5, 7) is divided by the line x + y = 4?
19. By using the concept of equation of a line, prove that the three points (-2, -2), (8, 2) and (3, 0) are collinear.
20. Find the equation of a straight line which passes through the point (4, -2) and whose intercept on y – axis is twice
that on x – axis.
21. Find the equation of the line which cuts off equal and positive intercepts from the axes and passes through the
point (α, β).
22. Find the equation of the line which passes through the point (3,4) and the sum of its intercepts on the axes is 14.
23. Point R (h, k) divides a line segment between the axes in the ratio 1 : 2. Find the equation of the line.
24. Find the equation of the line whose perpendicular distance from the origin is 4 units and the angle which the
normal makes with the positive direction of x –axis is 15o
.
25. Find the value of θ and p, if the equation x cos θ + y sin θ = p is the normal form of the line
√3 x + y + 2 = 0

2. 26. Reduce the lines 3x – 4y + 4 = 0 and 4x – 3y + 12 = 0 to the normal form and hence determine which line is
nearer to the origin.
27. Find the values of k which the line (k – 3) x – (4 – k2
)y + k2
– 7k + 6 = 0 is
(i) parallel to the x – axis.
(ii) parallel to the y – axis.
(iii) passing through the origin.
28. Find the value of λ, if the lines 3x – 4y – 13 = 0, 8x – 11y – 33 = 0 and 2x – 3y + λ = 0 are concurrent.
29. For what value of λ are the three lines 2x – 5y + 3 = 0, 5x – 9y + λ = 0 and x – 2y + 1 = 0 concurrent?
30. Find the equation of the line which is parallel to 3x – 2y + 5 = 0 and passes through the point (5, -6).
31. Find the coordinates of the foot of the perpendicular drawn from the point (1, -2) on the line y = 2x + 1.
32. Find the image of the point (-8, 12) with respect to the line mirror 4x + 7y + 13 = 0.
33. Find the equation of the perpendicular bisector of the line joining the points (1, 3) and (3, 1).
34. Find the image of the point (2, 1) with respect to the line mirror x + y – 5 = 0.
35. If the image of the point (2, 1) with respect to the line mirror be (5, 2), find the equation of the mirror.
36. Find the equation of a line drawn perpendicular to the line
𝑥
4
+
𝑦
6
= 1 through the point where it meets
the y – axis.
37. The line through (h, 3) and (4, 1) intersects the line 7x – 9y – 19 = 0 at right angle. Find the value of h.
38. Find the image of the point (3, 8) with respect to the line x + 3y = 7 assuming the line to be a plane mirror.
39. If the lines y = 3x + 1 and 2y = x + 3 are equally inclined to the line y = mx + 4. Find the values of m.
40. Find the slope of the lines which make an angle of 45o
with the line 3x – y + 5 = 0.
41. If p is the length of the perpendicular from the origin to the line
𝑥
𝑎
+
𝑦
𝑏
= 1,then prove that :
1
𝑝2
=
1
𝑎2
+
1
𝑏2
.
42. If p and p' be the perpendicular from the origin upon the straight lines x secθ + y cosecθ = a and
x cosθ - y sinθ = a cos 2θ. Prove that 4p2
+p'2
= a2
.
43. What are the points on x – axis whose perpendicular distance from the line 4x + 3y = 12 is 4?
44. Find the points on y – axis whose perpendicular distance from the line 4x – 3y – 12 = 0 is 3.
45. Show that the product of perpendiculars on the line
𝑥
𝑎
𝑐𝑜𝑠 𝜃 +
𝑦
𝑏
𝑠𝑖𝑛 𝜃 = 1 from the points (±√𝑎2 − 𝑏2 ,0) is b2
46. What are the points on y – axis whose distance from the line
𝑥
3
+
𝑦
4
= 1 is 4 units?
47. Find the perpendicular distance of the line joining the points (cos θ, sin θ) and (cos ϕ, sin ϕ) from the origin.
48. Show that the equation of the straight line through the origin making angle ϕ with the line y = mx+b
is
𝑦
𝑥
=
𝑚 ±𝑡𝑎𝑛 ∅
1 ∓𝑚 𝑡𝑎𝑛 ∅
.
49. Find the equation of the straight line which passes through the point (2, -3)and the point of intersection of the
lines x + y + 4 = 0 and 3x – y – 8 = 0.
50. Find the equation of the straight line which passes through the intersection of the lines x - y - 1 = 0 and
2x – 3y + 1 = 0 and is parallel to
(i) x- axis (ii) y- axis (iii) 3x + 4y = 14.
51. Find the equation of the straight line which passes through the intersection of the lines x + 2y = 5 and
3x + 7y = 17 and is perpendicular to the straight line 3x + 4y = 10.

3. CHAPTER – 11
CONIC SECTION
1. Find the equation of a circle whose centre is (2, -3) and radius 5.
2. Find the equation of a circle whose radius is 6 and the centre is at the origin.
3. Find the equation of the circle which passes through the point of intersection of the lines
3x – 2y – 1 = 0 and 4x + y – 27 = 0 and whose centre is (2, -3).
4. Find the centre and radius of each of the following circles :
(i) x2
+ (y +2)2
= 9
(ii) x2
+ y2
– 4x + 6y = 12
(iii) (x + 1)2
+ (y – 1)2
= 4
(iv) x2
+ y2
+ 6x - 6y + 4 = 0
5. Find the centre and radius of the circle given by the equation 2x2
+ 2y2
+ 3x + 4y +
9
8
= 0.
6. Find the equation of the circle concentric with the circle 2x2
+ 2y2
+ 8x + 10y – 39 = 0 and having its area
equal to 16π square units.
7. Find the equation of the circle that passes through the points (1, 0), (-1, 0) and (0, 1).
8. Find the equation of the circle which passes through the points (5, -8), (2, -9) and (2, 1). Find also the
coordinates of its centre and radius.
9. Show that the points (9, 1), (7, 9) (-2, 12) and (6, 10) are concyclic.
10. Find the equation of the circle which passes through the points (1, -2) and (4, -3) and its centre on the
line 3x + 4y = 7.
11. Find the equation of the circle passing through the points :
(i) (5, 7), (8, 1) and (1, 3)
(ii) (1,2), (3, -4) and (5, -6)
(iii) (5, -8), (-2, 9) and (2, 1)
(iv) (0, 0), (-2, 1) and (-3, 2)
12. Find the equation of the circle which passes through (3, -2), (-2, 0) and has its centre on the
line 2x – y = 3.
13. Find the equation of the circle which passes through the points (3, 7), (5, 5) and has its centre on the
line x – 4y = 1.
14. Show that the points (5, 5), (6, 4) (-2, 4) and (7, 1) all lie on a circle, and find its equation, centre and radius.
15. Show that the points (3, -2), (1, 0) (-1, -2) and (1, -4) are concyclic.
16. Prove that the radii of the circles x2
+ y2
= 1, x2
+ y2
– 2x - 6y - 6 = 0 and x2
+ y2
– 4x - 12y - 9 = 0 are in
A.P.
17. Foe the following parabolas find the coordinates of the foci, the equations of the directrices and the lengths of
the latus rectum :
(i) y2
= 8x (ii) x2
= 6y (iii) y2
= -12x (iv) x2
= -16y
18. if a parabola reflector is 20 cm in diameter and 5 cm deep, find its focus.
19. An arc is in the form of a parabola with its axis vertical. The arc is 10 m high and 5m wide at the base. How
wide is it 2 m from the vertex of the parabola.

4. 20. For the following ellipse find the lengths of major and minor axes,coordinates of foci and vertices , and the
eccentricity :
(i) 16x2
+ 25y2
= 400
(ii) 3x2
+ 2y2
= 6
21. Find the equation of the ellipse whose axes are along the coordinate axes,vertices are (±5,0) and foci at
(± 4, 0) .
22. Find the equation of the ellipse whose axes are along the coordinate axes,foci at ( 0,± 4) and eccentricity
4/5.
23. An arc is in the form of a semi – ellipse. It is 8m wide and 2 m high at the centre. Find the height of the arch
at a point 1.5 m from one end.
24. Find the equation of an ellipse whose foci are at (±3, 0) and which passes through (4, 1).
25. Find the equation of an ellipse whose axes lie along coordinate axes and which passes through
(4,3) and (-1, 4).
26. Find the equation of the ellipse whose vertices are ( 0,± 10) and eccentricity e =
4
5
.
27. For the following hyperbolas find the lengths of transverse and conjugate axes,eccentricity and coordinates
of foci and vertices; length of the latus – rectum, equations of the directories :
(iii) 16x2
- y2
= 144 (ii) 3x2
- 6y2
= -18
28. Find the equation of the hyperbola, referred to its principal axes as axes of coordinates, in the following cases
(i) Vertices at (±5,0), Foci at (±7,0)
(ii) Vertices at (0,±5) , e =
4
3
29. Referred to its principal axes as axes of coordinates find the equation of the hyperbola whose foci are at
(0, ± √10) and which passes through the point (2,3).
30. Find the axes, eccentricity, latus – rectum and the coordinates of the foci of the hyperbola
25x2
- 36y2
= 225.
31. In each of the following find the equations of the hyperbola satisfying the given conditions :
(i) Vertices at (0 ,±3), Foci at (0,±5)
(ii) Vertices at (±7, 0), e =
4
3
(iii) Foci at (0,±12) , latus – rectum = 36.