Question

If the circles x² + y² = a and x² + y² − 6x − 8y + 9 = 0, touch externally, then a =
(a) 1
(b) −1
(c) 21
(d) 16

Answer 1

(a) 1

x² + y² = a ...(1)
And, x² + y² − 6x − 8y + 9 = 0 ...(2)



Let circles (1) and (2) touch each other at point P.

The centre of the circle x² + y² = a, O, is (0, 0).

The centre of the circle x² + y² − 6x − 8y + 9 = 0, C₁, is (3, 4).

Also, radius of circle (1) = $\sqrt{a}$ = OP

Radius of circle (2) = $\sqrt{9+16-9}=4$ = C₁P

From figure, we have:

$$\begin{gathered} C_{\mathit{1}}O\mathit{=}{C}_{\mathit{1}}P\mathit{+}OP \\ \mathit{\Rightarrow }\sqrt{{\mathit{3}}^{\mathit{2}}\mathit{+}{\mathit{4}}^{\mathit{2}}}\mathit{=}4+\sqrt{\mathrm{a}} \\ \mathit{\Rightarrow }5\mathit{=}4+\sqrt{\mathrm{a}} \\ \mathit{\Rightarrow }a\mathit{=}1 \end{gathered}$$