Question

If the distances from the origin to the centres of three circles $x ^ { 2 } + y ^ { 2 } + 2 \lambda _ { i } x - c ^ { 2 } = 0 ( i = 1,2,3 )$ are in G.P. then the lengths of the tangents drawn to them from any point on the circle $x ^ { 2 } + y ^ { 2 } = c ^ { 2 }$ are in

• A. A.P.
• B. G.P.
• C. H.P.
• D. None of these

Answer 1

Answer: (B)

G.P.

Answer 2

The centre of the given circles are $\left( - \lambda _ { i } , 0 \right) ( i = 1,2,3 )$

The distances from the origin to the centre are $\lambda _ { i } ( i = 1,2,3 )$ It is given that

Let $P ( h , k )$ be any point on the circle $x ^ { 2 } + y ^ { 2 } = c ^ { 2 }$, then,$h ^ { 2 } + k ^ { 2 } = c ^ { 2 }$

Now, $L _ { i }$ = length of the tangent from (h, k) to

$x ^ { 2 } + y ^ { 2 } + 2 \lambda _ { i } x - c ^ { 2 } = 0$ $= \sqrt { h ^ { 2 } + k ^ { 2 } + 2 \lambda _ { i } h - c ^ { 2 } }$

=$\sqrt { c ^ { 2 } + 2 \lambda _ { i } h - c ^ { 2 } }$ $\left[ \because h ^ { 2 } + k ^ { 2 } = c ^ { 2 } \right.$ and $\left. i = 1,2,3 \right]$

Therefore, $L _ { 2 } ^ { 2 } = 2 \lambda _ { 2 } h = 2 h \left( \sqrt { \lambda _ { 1 } \lambda _ { 3 } } \right)$

$= \sqrt { 2 h \lambda _ { 1 } } \sqrt { 2 h \lambda _ { 3 } } = L _ { 1 } L _ { 3 }$. Hence, $L _ { 1 } , L _ { 2 } , L _ { 3 }$ are in G.P