First slide

Definition of a circle

Question

if the distances from the origin of the centres of
three circles $x^{2} + y^{2} + 2 λ_{i} x - c^{2} = 0 (i = 1, 2, 3)$ are in GP,
then the lengths of the tangents drawn to them from any point on the circle
$x^{2} + y^{2} = c^{2}$ are in

Moderate

A

A.P.
B

G.P.
C

H.P.
D

none of these

Solution

The centres of the given circles are $(- λ_{i}, 0)$
$(i = 1, 2, 3)$ The distances from the origin of the centres are
$λ_{1}, λ_{2}$ and $λ_{3} .$
It is given that $λ_{2}^{2} = λ_{1} λ_{3}$
Let $P (h, k)$ be any point on the circle $x^{2} + y^{2} = c$ .Then
$h^{2} + k^{2} = c^{2}$
Now,
$L_{i} =$ Length of the tangent from $(h, k)$ to
$\begin{array}{l} x^{2} + y^{2} + 2 λ_{i} x x - c^{2} = 0 \\ \Rightarrow L_{i} = \sqrt{h^{2} + k^{2} + 2 λ_{i} h - c^{2}} \\ \Rightarrow L_{i} = \sqrt{c^{2} + 2 λ_{i} h - c^{2}} \\ [∵ h^{2} + k^{2} = c^{2}] \\ \Rightarrow L_{i} = \sqrt{2 λ_{j} h}, \\ i = 1, 2, 3 . \\ ∴ {L_{2}}^{2} = 2 λ_{2} h \\ \Rightarrow {L_{2}}^{2} = 2 h (\sqrt{λ_{1}} λ_{3}) \\ \Rightarrow L_{2}^{2} = \sqrt{2 λ_{1} h} \sqrt{2 λ_{3} h} = L_{1} L_{3} \\ [∵ {λ_{2}}^{2} = λ_{1} λ_{3}] \end{array}$
Hence, $L_{1}, L_{2}, L_{3}$ are in G.P

Get Instant Solutions

When in doubt download our app. Now available Google Play Store- Doubts App

Recieve an sms with download link

Doubts App

OR

SCAN ME

Doubts App

Doubts App