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Number of Common Tangents to Two Circles in Different Conditions

The common ch...

Question

The common chord of the circle $x^{2} + y^{2} + 6 x + 8 y - 7 = 0$ and another circle passing through origin, which touches the line y = x, always passes through the point $(α, β)$ , then $| α - β |$ =___

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Solution

$x^{2} + y^{2} + 6 x + 8 y - 7 = 0$ ---------------------- (1)
$x^{2} + y^{2} + 2 g x + 2 g y = 0$ ---------------------- (2)
Equ of common chord is
2(g - 3) x + 2 (-g -4) y + 7 = 0
$\Rightarrow$ (-6x-8y+7)+g (2x-2y)=0
Always P. T. $(\frac{1}{2}, \frac{1}{2})$

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