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If the chord of contact of the point $P (h, k)$ with respect to the circle $x^{2} + y^{2} - 4 x - 4 y + 8 = 0$ meets the circle in two distinct points and it also makes an angle $45^{\circ}$ with the positive X-axis in the positive direction,then (h,k) cannot be

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$(\frac{5}{2}, \frac{3}{2})$

$(2, 2)$

$(\frac{5}{3}, \frac{7}{3})$

$(3, 1)$

answer is D.

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Detailed Solution

Equation of chord at $P (h, k)$ with respect to the circle $x^{2} + y^{2} - 4 x - 4 y + 8 = 0$ is $S_{1} = 0$

chord of contact is $h x + k y - 2 (x + h) - 2 (y + k) + 8 = 0$

$h x + k y - 2 x - 2 h - 2 y - 2 k + 8 = 0$

$(h - 2) x + (k - 2) y + (- 2 h - 2 k + 8) = 0$

by verification option D is saticefied

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