First slide

Definition of a circle

Question

The centre of the circle passing through (0, 0) and (1, 0) and touching the circle $x^{2} + y^{2} = 9$ ,is

Moderate

A

( 3/2, 1/2)
B

(1/2, 3/2)
C

(1/2, 1/2)
D

$(1 / 2, \pm \sqrt{2})$

Solution

Let the equation of the circle be
$x^{2} + y^{2} + 2 g x + 2 f y + c = 0$
This passes through (0, 0) and (1, 0).
$c = 0 and 1 + 2 g + c = 0 \Rightarrow c = 0, g = - 1 / 2$
The circle $x^{2} + y^{2} + 2 g x + 2 f y + c = 0$ touches the circle $x^{2} + y^{2} = 9 .$
$\begin{array}{l} \sqrt{g^{2} + f^{2}} = 3 \pm \sqrt{g^{2} + f^{2} - c} [Using C_{1} C_{2} = r_{1} \pm r_{2}] \\ \sqrt{1 / 4 + f^{2}} = 3 \pm \sqrt{1 / 4 + f^{2}} \\ 3 = 2 \sqrt{1 / 4 + f^{2}} \Rightarrow f^{2} = 9 / 4 - 1 / 4 = 2 \Rightarrow f = \pm \sqrt{2} \end{array}$

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