Chapter 10. Circles

Class 10 Circles Multiple Choice Questions and Solutions:

Question :  A circle can have …………… parallel tangent(s) at the most . [SEBA 2019 ]

(a) 1          (b)  2           (c)  3          (d)  4

Solution : (b) 2

[ A circle can have two parallel tangent(s) at the most . ]

Question : A tangent PQ at a point P of a circle of radius 5 cm meets a line through the centre O at a point Q so that  . Length PQ is :

(a) 12 cm      (b) 13 cm     (c) 8.5 cm    (d)  cm

Solution:  (D)   cm .

[ In figure :

  

In   , we have             

  

 

  ]  

Question :  In the circle given in figure, the number of tangents parallel to tangent PQ is :    [CBSE 2020 basic]

       

 (a) 0            (b) 1            (c) 2          (d) many

Solution:  (a)  1

Question :  From a point Q, the length of the tangent to a circle is 24cm and the distance of Q from the centre is 25cm .The radius of the circle is

(a)  7cm      (b)  12cm      (c) 15cm     (d) 24.5cm

Solution:  (a)  7 cm

[ In figure :

 

Here , OQ = 25 cm and  QT = 24 cm  

In OQT , we have

  

 

 cm   ]                                             

Question :  In fig. 10.11,if TP and TQ are the two tangents to a circle with centre O so that  then is equal to :

(a)  60°        (b)  70°      (c)  80°      (d)  90°

Solution:  (b)  70°

[  In figure : 

   

Here, and

Since,  OPTQ is cyclic quadrilateral , We have

 

 

 

  ]

Question : If the tangents PA and PB from a point P to a circle with centre O are inclined to each other  at an angle of 80°, then  is equal to

(a)  50°     (b) 60°       (c)  70°         (d) 80°

Solution:   (a) 50°

[ In figure :

  

Since, OP is the angle bisector of  .

So,

In , we have

   

 

 

]

Question :  In figure, from an external point P, two tangents PQ and PR are drawn to a circle of radius 4 cm with centre O . If , then length of PQ is :     [CBSE 2020 standard]

   

   (a) 3 cm         (b) 4 cm        (c) 2 cm       (d)  cm

Solution:    (b) 4 cm                

[ Since  OP is the angle bisector of  .

 

 In  we have ,

  

 

     

  So, OPQ is an equilateral triangle .

    cm           ]

Question : In figure, AB is a chord of the circle and AOC is its diameter such that . If AT is the tangent to the circle at the point A , then is equal to :               

   

 (a) 65°            (b) 60°          (c) 50°          (d) 40°    

Solution:     (c)  50°   

[ Since AC perpendicular to AT .

   and (Angle in a semicircle is a right angle)

  In  we have ,                 

 So ,   

   

 But ,  

    ]

Question : From an external point P, tangents PA and PB are drawn to a circle with centre O . If   , then   is :

 

 (a)  30°        (b)  35°       (c)   45°      (d)  25°  

Solution:     (d)  25°    .

 [ Since APB is an equilateral triangle .

 

    [    ]

   

   

  But  

          ]

Question :  In figure, if O is the centre of a circle, PQ is a chord and  the tangent PR at P makes an angle of 50°  with PQ , then is equal to :        

  

 (a) 100°       (b) 80°        (c) 75°      (d) 90°  

Solution:    (a) 100°  

[  Since OP is perpendicular to PR .

 

  

    POQ is an equilateral triangle .            

 

  

     ]

Question :  In figure, PQ is tangent to the circle with centre at O , at the point B .If , then is equal to :  [ CBSE 2020]

     

 (a) 50°        (b)  60°       (c) 40°      (d) 80°

Solution:   (a)  50°

[ Since AOB is an isosceles triangle .

In, we have

But 

 

]

Class 10 Maths Chapter 10 . Circles 3 and 4 Marks Questions and Solutions: 

Question: Prove that the lengths  of tangents drawn from an external point to a circle are equal . [SEBA 2015 , 2017]

Solution:   Given O be a centre of the circle and a point P lying outside the circle and two tangents PQ , PR on the circle from P .

 To prove :    

Construction : We join ,  and  .

 Proof : In figure ,

In  and    we have , 

   ( Radius of the same circle)                                       

  ( Common)              

    [ OAAP and OBBP]

 [  R.H.S rule]

    (C.P.C.P.)   Proved.

Question : Prove that the tangent at any point of a circle is perpendicular to the radius through the point of contact .  

Solution:  Given, a circle with centre O and a tangents XY to the circle at a point P .

 To prove :  OP is perpendicular to XY .

   Construction : we draw a point Q on XY other than P and join OQ .

  Proof :  The point Q must lie outside the circle .

  Therefore , OQ is longer than the radius OP of the circle . 

        i.e. , OQ > OP

   

 Every point on the line XY except the point P , OP is the shorter of all the distance of the point O to the points of XY .

So, OP is perpendicular to XY .

Question : Prove that in two concentric circles , the chord of the larger circle , which touches the smallest circle , is bisected at the point of contact . [SEBA 2023]

Solution : Given O be a centre of the concentric circle and  respectively  and a chord AB of the larger circle  which touches the smaller circle  at the point P .

To prove :

Construction : We join OP.

Proof : Since , the tangent at any point of a circle is perpendicular to the radius through the point of contact .

     

So,  AB is a tangent to  at P and OP is its radius.

Therefore, 

Again, AB is a chord of the circle   and  .

 Therefore, OP is the bisector of the chord AB .

So,  AP = BP  Proved .

Question :  Prove that the tangents drawn at the ends of a diameter of a circle are parallel. [SEBA 2019 , 2021]

Solution:  let AB and CD are two tangent touch at X and Y of the diameter XY of the circle with centre O .

To prove:   .

Proof : Since, the tangent to a circle is perpendicular to the radius through the point of contact .

    

 

and

 

So,  and XY is a transversal .

   Proved .

Question :  A quadrilateral  ABCD is drawn to circumscribe  a circle a circle (see Fig. 10.12). Prove that

Solution:  Let ABCD be a quadrilateral with sides AB, BC, CD, and AD, and let P, Q, R and S be the points where the circle with centre O touches the sides, respectively.

To prove  :  .

Proof :  Since the length of the two tangents from an external point to a circle are equal .

    

   (From A)…….(i)  

     (From B)…….(ii)

   (From C)…….(iii) 

     (From D )…….(iv) 

  Addin (i) ,(ii) ,(iii) and (iv) , we have

 

    Proved.

Question :   Two tangents TP and TQ are drawn to a circle with centre O from an external point T . Prove that  . [SEBA 2018]

Solution:  Given, O be a centre of a circle and an external point T and two tangents and  to the circle , where P and Q are the points of contact  .

To prove :    

Proof :  Since the length of tangents drawn from an external point to a circle are equal .

So,  

  

  In   , we have 

    

So, PTQ is an isosceles triangle .

    

But    [   OPPT ]

  

 

  In  we have ,

   

 

   

       [ From  ]

 

     Proved

Question : Prove that the parallelogram circumscribing a circle is a rhombus . [SEBA 2015 , 2020]

Solution:  Given   be a parallelogram with sides AB, BC, CD, and AD, and let P, Q, R and S be the points where the circle with centre O touches the sides, respectively.

To Prove :   is a rhombus .

Proof : Since the lengths of tangents drawn from an external point to a circle are equal .

  

  Therefore ,    (From A )……. 

      (From B )…….

     (From C )……. 

      (From D )…….  

   

    

Since  be a parallelogram .

Thus,  and  

 

 

       

 Therefore,   is a rhombus.     Proved .

Question :   In Fig. 10.13, XY and    are two parallel tangents to a circle with centre O and another tangent AB with  point of contact C intersecting XY at A and  X’Y’ at B .Prove that   . [SEBA 2016]

Solution:  Given , and ' are two parallel tangents to a circle with centre O and another tangent AB with point of contact C intersecting  at A and ' at B . To Prove  

Construction :  Join  and .

Proof :  Since the lengths of the two tangents from an external point to a circle are equal and also the subtend equal angles at the centre .

   

Therefore , is a bisector of   

and is a bisect of  

Since,  and  is a transversal 

 

[From  and  ]

In   we have,

 

   [ From  ]

  Proved.

Question :  A circle is inscribe in a   having sides 8 cm , 10 cm , 12 cm as shown in the following figure . Find AD , BE and CF . [SEBA 2017]

Solution : Here, and

Since, the lengths of tangents drawn from an external point to a circle are equal .

So , and

 

Let , , and

We have, 

  

 

 

 ,and

Therefore, ,  and