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General Equation of a Circle

Find the equa...

Question

Find the equation of the circles which touch at the point (1,2) , one of the two intersecting lines which are tangents to it and whose equations are $7 x - y - 5 = 0 a n d x + y + 13 = 0$

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Solution

Centre must lie on the bisector of the angles between the lines which are found to be
$x - 3 y = 35 a n d 3 x + y = - 15$
If (h,k) be the centre , then
$h - 3 k = 35$ ......(1)
and $3 h + k = - 15$ ......(2)
$C B ⊥ B P \Rightarrow \frac{k - 2}{h - 1} \times 7 = - 1$
or $h + 7 k - 15 = 0$ ......(3)
Solving (1), (3) and (2), (3) , we get the centres as
$C_{1} (- 6, 3) a n d C_{2} (29, - 2)$
$r_{1}^{2} = 50 a n d r_{2}^{2} = 800$
$∴$ Circle are $(x + 6)^{2} + (y - 3)^{2} = 50$
and $(x - 29)^{2} + (y + 2)^{2} = 800$

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