Find the equation of a circle whose centre is (3, -1) and which cuts off a chord of length 6 units on the line $2 x - 5 y + 18 = 0$ .

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Solution

Distance from centre O $(3, - 1)$ to the line $2 x - 5 y + 18 = 0$ is $O E = \frac{6 + 5 + 18}{\sqrt{2^{2} + (- 5)^{2}}}$
$= \frac{29}{\sqrt{4 + 25}}$
$∴ O E = \sqrt{29}$
From triangle $Δ O E A$
$A O^{2} = O E^{2} + A E^{2}$
$= 9 + 29$
$∴ A O = \sqrt{38}$ , which is the radius of the circle.
Equation of circle with centre (3, -1) and radius as $\sqrt{38}$ is
$(x - 3)^{2} + (y + 1)^{2} = (\sqrt{38})^{2}$
$x^{2} - 6 x + 9 + y^{2} + 2 y + 1 = 38$
$x^{2} + y^{2} - 6 x + 2 y = 28$

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Find the equation of the circle whose centre is (3, -1) and which cut-off an intercept of length 6 from the line $2 x - 5 y + 18 = 0$ .

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