A normal is drawn to the parabola $y^{2} = 9 x$ at the point $P (4, 6)$ . A circle is describled on $S P$ as diameter, where $S$ is the focus . The length of the intercept made by the circle on the normal at a point $P$ is $\frac{m}{n}$ , where $m, n$ are integers and $H C F (m, n) = 2$ , then

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$m + n = 19$

$m - n = 22$

$m - n = 11$

$m + n = 38$

answer is A, C.

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Detailed Solution

Equatio nof the circle with $S P$ as diameter

$(x - 4) (x - \frac{9}{4}) + y (y - 6) = 0$

The center is $(\frac{25}{8}, 3)$ and Radius is $\frac{25}{8}$

Equation of normal at $P (4, 6)$ is $4 x + 3 y = 34$

Length of the chord is $2 \sqrt{r^{2} - d^{2}} = \frac{15}{4}$

Hence, $m + n = 19, m - n = 11$

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