MATHEMATICS
CLASS X
CIRCLE
MISCELLANEOUS EXERCISE
1. In fig 1.1, AB is a diameter of a circle C(O, r) and radius OD AB.If C is a point on arc DB. Find
2. In fig 1.2, O is the centre of the circle. Calculate
(i)
3. In fig. 1.3, O is the centre of the circle and AOB = 70°, Find
4. In fig 1.4, O is the centre of the circle. Determine.
(i)
5. In fig . 1.5, given that AB = AC and Determine
6. In fig. 1.6, O is the centre of the circle. Determine
7. Two chords AB and CD intersect inside a circle at point O, Prove that
8. Two lines OAB and OCD drawn from an exterior point O to a circle intersect the circle at A and B, and at C and D respectively. Prove that
9. AB and XY are parallel chords of a circle. AY intersects BX at O. Prove OX = OY.
10. AB is a chord of a circle whose centre is O. If p is any point on the minor arc AB, prove that
11. In a quadrilateral ABCD, in which AB = AC = AD, show that See fig. 1.8.
12. In fig. 1.9, ABC, AEG and HEC are straight lines. Prove that are
supplementary.
13. In a circle with centre O, chords AB and CD intersect inside the circumference at E. Prove that .
14. I is the in centre of ABC . AI when produced meets the circumcircle of ABC in D. If BAC = 66° and = 80°. Calculate [See fig 1.10]
(i) (ii) (iii) BID
15. Fill up the blanks with appropriate word.
(i) The set of all those points in a plane that ate at a given constant distance from a given fixed point is called a ……….
(ii) Circles having the same centre are called ……….
(iii) The line drawn from the centre of a circle, perpendicular to a chord … the chord.
(iv) The perpendicular bisector of a chord of a circle passes through the ………
(v) A line-segment joining two points on the circle passes through the ………
(vi) Two circles are said to be congruent, if and only if their …… are equal.
(vii) If two arcs of a circle are congruent, then the corresponding chords are ………
(viii) Equal chords cut off …… arcs.
(ix) Perpendicular bisectors of two non-parallel chords of a circle intersect each other at the ………… of the circle.
(x) If two circles intersect in two distinct points, the line joining their centres is ………… of their common chord.
(xi) In a right triangle, the circumcentre is the ………… of the hypotenuse.
(xii) For an equilateral triangle, at the circumcentre coincides with …………
(xiii) If the semicircle drawn with one side of a triangle as diameter passes through the opposite vertex, then the measure of the angle at the vertex is ……….
(xiv) If opposite angles of a quadrilateral are supplementary, then the quadrilateral is called …….
(xv) There non-collinear points describe a ………. circle.
16. In Fig. 1.11, O is the centre of the circle and BC = AO. Which of the following relationship between x and y is correct?
Ax = 3y always
Bx = 2y always
Cx = 4y always
Dx = 2y or x = 3y.
17. In fig 1.12, is an isosceles triangle with AB = AC and Find
18. With reference to Fig. 1.14 determine angles a, b and c.
19. ABCD is a cyclic quadrilateral. AE is drawn parallel to CD and BA is produced to F. If
(See fig. 1.15)
20. ABCD is a cyclic quadrilateral. AE is drawn parallel to CD and BA is produced to F. If
21. If two sides of a cyclic quadrilateral are parallel, prove that (i) the remaining two sides are equal and (ii) both the diagonals are equal.
22. In a cyclic quadrilateral ABCD, if
23. If the two sides of a pair of opposite sides of a cyclic quadrilateral are equal, prove that its diagonals are equal.
24. ABCD is a cyclic quadrilateral, in which AB and CD when produced meet in E and EA = ED. Prove that
(i) AD || BC (ii) EB = EC.
25. Circles are described on the sides of a triangle as diameters. Prove that the circles on any two sides intersect each other on the third side (or third side produced).
26. Prove that the circles described on the our sides of a rhombus as diameter, pass through the point of intersection of its diagonals.
27. Prove that the perpendicular bisectors of the sides of a cyclic quadrilateral are concurrent.
28. Prove that the centre of the circle circumscribing the cyclic rectangle ABCD is the point of intersection of its diagonals.
29. The diagonals of a cyclic quadrilateral are at right angles. Prove that the perpendicular from the point of their intersection on any side, when produced backward bisects the opposite side.
30. ABCD is a cyclic quadrilateral AB and DC are produced to meet in E. Prove that
31. In fig.1.16, AB is a diameter of the circle. Also, AD Prove that CE = AD.
32. In Fig. 1.17, AB = AD = PB and Determine (i)
Hence, or otherwise, prove that AP is a parallel to DB.
33. AB and CD are two parallel chords of a circle, which are on opposite sides of the centre, such that AB = 10 cm, CD = 24 cm and the distance between AB and CD is 17 cm. Find the radius of the circle
34. In in fig. 1.18, is a right angle. A semicircle is drawn on AB as diameter. P is any point of AC produced. When joined, BP meets the semicircle in point D. Prove that
AB² = AC.AP + BD.BP.
35. X and Y are centres of circles of radius 9 cm and 2 cm and XY = 17 cm. Z is the centre of a circle of radius r cm. which touches the above circles externally. Given that write an equation in r and solve it for r.
36. is right angled at B. On the side AC, a point D is taken such that AD = DC and AB = BD. Find the measure of
37. In fig. 1.20. O is the centre of the circle. Determine (i)
38. AB is a chord of a circle, whose centre is O. If p is any point on the mirror arc AB, prove that (See Fig. 1.20)
39. ABCD is a rectangle, Prove that the centre of the circle through A, B, C, D is the point of intersection of its diagonals.
40. In Fig. 1.22, AB = CD. Prove that BE = DE and AE = CE. Where E is the point of intersection of AD and BC.
41. In fig. 1.23, a diameter AB of a circle bisects a chord PQ. If AQ || PB, prove that chord PQ is also a diameter of the circle.
42. Two circles are drawn with sides AB and AC of a triangle ABC as diameters. The circles intersect at a point D. Prove that D lies on BC.
43. Two diameters of a circles intersect each other at right angles. Prove that the quadrilateral formed by joining their end points is a square.
44. ABCD is a cyclic quadrilateral with AD || BC. Prove that AB = DC.
45. Prove that bisectors of the sides of a cyclic quadrilateral are concurrent.
46. Prove that the circle drawn with any side of a rhombus as a diameter, passes through the point of intersection of its diagonals.
47. D is the mid point of side BC of an isosceles triangle ABC with AB = AC. Prove that the circle drawn with either of the equal sides as a diameter passes through the point D.
48. The circle passing through the vertices A, B and C of a parallelogram ABCD intersects side CD (or CD produced) at the point P. Prove that AP = AD.
49. Prove that the quadrilateral formed by angle bisectors of a cyclic quadrilateral is also cyclic.
50. D and E are respectively, the points on equal sides AB and AC of an isosceles triangle ABC such that AD = AE. Prove that the points B, C, E and D are concylic.
51. The diagonals AC and BD of a cyclic quadrilateral ABCD intersect at right angles at E. A line l through E and perpendicular to AB meets CD at F. Prove that F is the mid-point of CD. (see Fig. 1.24)
(i) AD.BC = AE.DB (ii) AB.DE = CE.DB.
MATHEMATICS
CLASS X
TANGENTS TO A CIRCLE
MISCELLANEOUS EXERCISE
1. AB is a chord of a circle with center O. The tangent at B meets Ao produced P. If
2. In given Fig. 2.1, PT is a tangent to a circle. If and m
3. Two Chords AB and CD of a circle intersect each other at O internally. If AO = 3.5 cm, CO = 5 cm and DO = 7 cm, find OB.
4. Two chords AB and CD of a circle intersect each other at P outside the circle. If AB = 5 cm, BP = 3 cm and PD = 7 cm, find CD.
5. Find the length of the tangent drawn from a point whose distance from the centre of the circle of the circle is 25 cm, Given that the radius of the circle is 7 cm.
6. In given fig. 2.2, to the circle intersect each other at A and B. The common tangne meet the two circles at C and D. Prove that
.
7. If PA and PB are tangents from an outside point P such that PA = 10 cm and Find the length of chord AB.
[ Hint : PA = PB therefore
8. In given Fig. 2.3, circles C(O, r) and C (O’, r/2) touch internally at a point A and AB is a chord of the circle C (O, r) intersecting C(O’, r/2) at C. Prove that AC = CB.
9. In a right the perpendicular BD on the hypotenuse AC is drawn. Prove that
(i) AC × AD = AB²
(ii) AC × CD = BC²
10. Two circles touch internally at a point P and a chord AB to the circle of larger radius intersects the other circles at C and D. Prove that
11. Two circles C(O, r) and C(O’, s) touch each other at P, externally or internally. A line is drawn to pass through P intersecting the two circles at Q and R respectively. Prove that OQ || O’R.
12. In the given fig. 2.4, the diameters of two wheels have measures 2 cm and 4 cm. Determine the lengths of the belts AD and BC that pass around the wheels if it is given that belts cross each other at right angles. [Hint : Joint OO’. If P is the point of intersection of belt calculate PA, similarly calculate PD. Also AD = CB]
13. In the given fig. 2.5. Find x if
14. If PAB is a secant to a circle intersecting the circle at A and B and PT is a tangent. Prove that PA × PB = PT². On the above theorem prove the following. Two circles intersect each other at P and Q. From a point R on PQ produced, two tangents RB and RC are drawn to the two circles touchng then at B and C. Prove that RB = RC.
15. AB and CD are two chords which when produced meet at P and if AP = CP, show that AB = CD.
(see fig 2.6)
MATHEMATICS
CLASS X
LINEAR EQUATIONS IN TWO VARIABLES
MISCELLANEOUS EXERCISE
Solve each of the following (from 1 to 5)
1. 3x – 5y + 1 = 0
x – y + 1 = 0
2. 3.
4.
5. Solve for u and v:
3(2u + v) = 7 uv
3(u + 3v) = 11 uv.
6. Solve :
and hence find a for which y = ax – 4.
7. Solve graphically the following system of equation :
2x + y – 3 = 0, 2x – 3y – 7 = 0
8. Determine graphically the coordinates of the vertices of a triangle the equations of whose sides are y = x, y = 2x, x + y = 6.
9. Show graphically that the following system of equations has infinitely many solutions :
2y = 4x – 6, 2x = y + 3
10. Show graphically that the following system of equations in inconsistent :
3x – 5y = 20
6x – 10y = – 40
11. Solve graphically the following system of linear equations :
2x – y = 2
4x – y = 8
Also, find the coordinates of the points where the lines meet the axis of x.
12. Show that the following system of equations has no solution :
x – 2y = 6
3x – 6y = 0
13. For what value of K, the following equations are in consistent :
x – 4y = 6
3x + ky = 5
14. Determine the value of for which the following system of equations has infinitely many solutions:
x + 3y = – 3
12 x + y =
15. Find the value of K for which the system of equations :
8 x + 5y = 9
Kx + 10 y = 15 has no solution.
16. There are two examination rooms A and B. If 10 candidates are sent from A to B, the number of students in each room is the same. If 20 candidates are sent from B to A the number of students A is double the number of students in B. Find the number of students in each rooms.