A pair of straight lines through $A (2, 7)$ is drawn to intersect the line $x + y = 5$ at $C$ and $D .$ If angle between the pair of straight lines is $\frac{π}{2},$ then the locus of incentre of $△ A C D$ is

2 x y - 6 x + 4 y = 28

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x y - 6 x + 4 y = 29

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x^{2} + y^{2} - 6 x + 4 y = 29

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x y - 6 x + 4 y = 28

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Solution

The correct option is A $2 x y - 6 x + 4 y = 28$

Let inradius be $r = O N = O M$
Since $∠ O A N = \frac{π}{4},$
$\Rightarrow A N = O N$
$∴ O A = \sqrt{2} r$
Let incentre be $O (h, k)$
$r = \frac{O A}{\sqrt{2}} = O M$
$\Rightarrow \frac{\sqrt{{(k - 7)}^{2} + {(h - 2)}^{2}}}{\sqrt{2}} = ∣ ∣ ∣ \frac{k + h - 5}{\sqrt{2}} ∣ ∣ ∣ \Rightarrow \sqrt{{(k - 7)}^{2} + {(h - 2)}^{2}} = | k + h - 5 | \Rightarrow {(k - 7)}^{2} + {(h - 2)}^{2} = {| k + h - 5 |}^{2} \Rightarrow - 14 k + 49 - 4 h + 4 = 25 + 2 k h - 10 h - 10 k \Rightarrow 2 k h + 4 k - 6 h - 28 = 0$

Hence, locus is $2 x y - 6 x + 4 y = 28$

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