Let y=x be the equation of a chord of the circle ? (in the closed half-plane) of diameter 10 passing through the origin. Let be another circle described on the given chord as its diameter. If the equation of the chord of the circle , which passes throug

Question

Let $y = x$ be the equation of a chord of the circle $C_{1}$ (in the closed half-plane $x \geq 0$ ) of diameter 10 passing through the origin. Let $C_{2}$ be another circle described on the given chord as its diameter. If the equation of the chord of the circle $C_{2}$ , which passes through the point (2,3) and is farthest from the center of $C_{2}$ , is $x + a y + b = 0$ , then a−b is equal to [2026]

Smart Achievers · Accepted Answer

(2)

$Equation of circle C_{2} is$

$x \geq 0$

$Its centre is (\frac{5}{2}, \frac{5}{2})$

$m_{A B} = - 1$

$∴$ $Slope of required chord = 1$

$∴$ $Equation of required chord is x - y + 1 = 0$

$∴$ $a = - 1, b = 2$

$∴$ $a - b = - 2$

Solution

1 10

2 -2

3 6

4 -6

Ans.
(2)

$Equation of circle C_{2} is$

$x^{2} + y^{2} - 5 x - 5 y = 0$

$Its centre is (\frac{5}{2}, \frac{5}{2})$

$m_{A B} = - 1$

$∴$ $Slope of required chord = 1$

$∴$ $Equation of required chord is x - y + 1 = 0$

$∴$ $a = - 1, b = 2$

$∴$ $a - b = - 2$

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