If $S : x^{2} + y^{2} = 9$ be a circle. $P A$ and $P B$ are pair of tangents on $S$ where $P$ is any point on the director circle of $S$ , then the radius of the smallest circle which touches $S$ externally and also the two tangents $P A$ and $P B$ is $α + β \sqrt{2}$ . Then the value of $α + β$ is:

Open in App

Solution

Director circle to $x^{2} + y^{2} = 9$ is $x^{2} + y^{2} = 18$
$\Rightarrow$ radius $= 3 \sqrt{2}$

$P$ is a point on director circle and $P A$ and $P B$ are pair of tangents to the circle $S$
$\Rightarrow P A ⊥ P B \Rightarrow P M = P N Q M ⊥ P A & Q N ⊥ P B$
$\Rightarrow P M Q N$ is a square.
$O R = 3$
Let $Q N = Q M = Q R = r O P = 3 \sqrt{2} P Q = r \sqrt{2}$
$O P = O R + R Q + Q P \Rightarrow 3 \sqrt{2} = 3 + r + r \sqrt{2} \Rightarrow r = \frac{3 (\sqrt{2} - 1)}{\sqrt{2} + 1} \Rightarrow r = 9 - 6 \sqrt{2} \Rightarrow r = α + β \sqrt{2} \Rightarrow α = 9, β = - 6 \Rightarrow α + β = 3$

Suggest Corrections

Similar questions

Q. If

C_{1} : x^{2} + y^{2} = (3 + 2 \sqrt{2})^{2}

be a circle and PA and PB are pair of tangents on

C_{1}

, where P is any point on the director circle of

C_{1}

, then the radius of smallest circle which touch

C_{1}

externally and also the two tangents

P A

and

P B

Two circles touch each other externally at a point C and P is a point on the common tangent at C. If PA and PB are tangents to the two circles, prove that PA = PB.

Q. In fig.,

P A

and

P B

are two tangents drawn from an external point

P

to a circle with centre C and radius

4 c m

. If

P A

⊥

PB, then the length of each tangent is:

From an external point P, tangents PA and PB are drawn to a circle with centre O. At one point E on the circle tangent is drawn, which intersects PA and PB at C and D respectively. If $P A = 14 c m$ , find the perimeter of $Δ P C D .$