CBSE Class 10 Mathematics Important Questions for Exam 2023 (Chapter - Circles)

 

Important Mathematics Questions of  Class X Chapter {Circles} for CBSE Examination 2023

1. If radii of the two concentric circles are 15 cm and 17 cm, then find the length of chord of one circle which is tangent to the other.  

2.      If two tangents making an angle of 120⁰ with each other are drawn to a circle of radius 6cm, then      find the angle between the two radii, which are drawn at the points of contact to the tangents.

     3.      PQ is a chord of a circle and R is point on the minor arc. If PT is a tangent at point P such        that  ÐQPT = 60 then find <PRQ.

4.      If a tangent PQ at a point P of a circle of radius 5cm meets a line through the Centre O at a point Q such that OQ = 12 cm then find the length of PQ.

5.      From a point P, two tangents PA and PB are drawn to a circle C (o , r) . If OP =2r, then what is the type of APB.

6.      If the angle between two radii of a circle is 130, then find the angle between the tangents at the end of the radii.

7.      ABCD is a quadrilateral. A circle centered at O is inscribed in the quadrilateral. If AB = 7cm , BC = 4cm , CD = 5cm then find DA.

8.      In a ∆ABC , AB= 8 cm, BC = 6 cm,    Ð ABC = 90⁰, then find radius of the circle inscribed in the triangle.

9.  A point p is 13cm from the Centre of the circle. The length of the tangent drawn from P to the circle is 12cm. Find the radius of the circle.

10.      Two tangents PA and PB are drawn from an external point P to a circle with Centre O. Prove that OAPB is a cyclic quadrilateral.

11.      If PA and PB are two tangents drawn to a circle with Centre O , from an external point P such that PA=5cm and ÐAPB = 60,  then find the length of the chord AB.

12.      CP and CQ are tangents from an external point C to a circle with Centre O .AB is another tangent which touches the circle at R and intersects PC and QC at A and B respectively . If CP = 11cm and BR = 4cm, then find the length of BC.

13.       If all the sides of a parallelogram touch a circle, show that the parallelogram is a rhombus.

14.       Prove that the perpendicular at the point of contact to the tangent to a circle passes through the      Centre of the circle.

15.      If quadrilateral ABCD is drawn to circumscribe a circle then prove that AB+CD=AD+BC.

16.      Prove that the angle between the two tangents to a circle drawn from an external point is supplementary to the angle subtended by the line segment joining the points of contact to the Centre.

17.      AB is a chord of length 9.6cm of a circle with Centre O and radius 6cm.If the tangents at A and B intersect at point P then find the length PA.

18.      The encircle of a ∆ABC touches the sides BC, CA &AB at D,E and F respectively. If AB=AC, prove that BD=CD.

19.      Prove that the intercept of a tangent between two parallel tangents to a circle subtends a right angle at the Centre of the circle.

20.      PQ and PR are two tangents drawn to a circle with Centre O from an external point P. Prove that ÐQPR=2ÐOQR.

 21.      Prove that the length of tangents drawn from an external point to a circle are equal. Hence, find BC, if a circle is inscribed in a ABC touching AB,BC &CA at P,Q &R respectively, having AB=10cm, AR=7cm &RC=5cm.

22.      Prove that the tangent at any point of a circle is perpendicular to the radius through the point of contact. Using the above, do the following: If O is the Centre of two concentric circles, AB is a chord of the larger circle touching the smaller circle at C, then prove that AC=BC.

23.      A circle touches the side BC of a ∆ABC at a point P and touches AB and AC when produced, at Q & R respectively. Show that AQ=1/2 (perimeter of ∆ABC).

24.      From an external point P, a tangent PT and a line segment PAB is drawn to circle with centre O, ON is perpendicular to the chord AB. Prove that PA.PB=PN²-AN².

25.      If AB is a chord of a circle with centre O, AOC is diameter and AT is the tangent at the point A, then prove that ÐBAT=ÐACB.

26.      The tangent at a point C of a circle and diameter AB when extended intersect at P. If ÐPCA=110⁰ , find ÐCBA.

27.      If PA and PB are tangents from an external point P to the circle with Centre O, the find ÐAOP+ÐOPA.

28.      ABC is an isosceles triangle with AB=AC, circumscribed about a circle. Prove that the base is bisected by the point of contact.

29.      AB is  diameter of a circle with Centre O. If PA is tangent from an external point P to the circle with ÐPOB=115⁰ then find ÐOPA.

30.  PQ and PR are tangents from an external point P to a circle with Centre. If ÐRPQ=120⁰, Prove that OP=2PQ.

31.  If the common tangents AB and CD to two circles C(O,r) and C’(O’r’) intersect at E, then prove that AB=CD.

32.  If a, b, c are the sides of a right triangle where c is the hypotenuse , then prove that radius r of the circle touches the sides of the triangle is given by r= (a+b-c)/2.