Quantitative Aptitude > Geometry

QUESTION

The sum of the areas of two circles, which touch each other externally, is 153 . If the sum of their radii is 15, find the ratio of the larger to the smaller radius.

(1) 4

(2) 2

(3) 3

(4) none of these.

SOLUTION

Let the radii of the 2 circles be and , then = 15 (given)

and = (given)

= 153

solving we get, = 12 and = 3

ratio of the larger radius to the smaller one is 12 : 3 = 4 : 1 hence option (1) is the answer.

QUESTION

In the adjoining figure, points A, B, C and D lie on the circle. AD= 24 and BC = 12. what is the ratio of the area of CBE to that of the triangle ?

(1) 1:4

(2) 1:2

(3) 1:3

(4) data insufficient.

SOLUTION

In and , ∟ CBA = ∟ CDA.

( a chord of a circle subtends equal angel at all points on the circumference, lying in the same segment)

similarly ∟ BCD = ∟ BAD and ∟ BEC = ∟ AED

Therefore ( AAA similarity rule)

Now

Hence hence option b).

QUESTION

and are two concentric circles while BC and CD are tangents to at point P and respectively. AB to the circle , which of the following is true?

(1)

(2)

(3)

(4) data insufficient

 

SOLUTION

 

l ( cp) = l (cr) ( since tangents drawn from the same point are equal)

hence l(BC) = l( CD) ( points P and R are midpoints of BC and CD as OP is ┴BC and OR ┴ CD)

Therefore m( arc BXC) = m(arc CYD)
hence quadrilateral OPCR is a cyclic quadrilateral.

Hence answer is option c)

Hence m∟PCR =

m∟POR = 2m ∟PQR = (angle at center)

m∟OPC + m∟ORC =

Two circles of radius 3 units and 4 units are at some distance such that length of the transverse common tangents and the length of their direct common tangents are in the ratio 1: 2 . What is the distance between the centres of those circles?

(1) units

(2) units

(3)        8 units

(4)        cannot be determined .

SOLUTION

Let x = distance between the centres of the circles.

T= length of the transverse common tangent.

D = length of direct common tangent.
 

And = 4 units and = 3 units 

then, t = --------------- 1) 

also d =  

------------------- 2) 

=  

Since  

units 

hence

hence x =

from 1) and 2 ) we get

d = 2t 

d = 8 units

hence answer is option 2 ).