Question

Choose the correct answer from the alternatives given.
In the given figure. Tangents TQ and TP are drawn to the larger circle (center O) and tangents TP and TR are drawn to the smaller circle (center O’). Find TQ:TR?

$\left( a \right)1:1$
$\left( b \right)5:4$
$\left( c \right)8:7$
$\left( d \right)7:8$

Answer 1

Hint: In this particular question use the property that from a fixed point tangent drawn on the circle are of equal lengths so use this concept to reach the solution of the question.

Complete step by step answer:
It is given that in the given figure tangents drawn from the fixed point T on the larger circle having center O, are TQ and TP.
Now as we all know that from a fixed point tangent drawn on the circle are of equal lengths.
Therefore, TQ = TP.................... (1)
Now it is also given that in the given figure tangents drawn from the fixed point T on the smaller circle having center O’, are TP and TR.
So again use the property of a circle that from a fixed point tangent drawn on the circle are of equal lengths.
Therefore, TP = TR................ (2)
Now from equations (1) and (2) we have,
Therefore, TQ = TP = TR
Therefore, we can say that
TQ = TR
$ \Rightarrow \dfrac{{TQ}}{{TR}} = \dfrac{1}{1}$
So the ratio of tangents TQ and TR (i.e. TQ:TR) = 1 : 1.

So, the correct answer is “Option a”.

Note: Whenever we face such types of questions the key concept involved in this is the property of tangents on the circle from an external fixed point which is stated above, so apply this property on larger as well as on smaller circle and simplify as above, we will get the required answer.