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Let $3x – y – 8 = 0$ be the equation of tangent to a parabola at the point $(7, 13)$. If the focus of the parabola is at $(– 1, – 1)$, then the equation of its directrix is?

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    The answer given is x + 8y +19 = 0 – 2017-02-10

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HINT:

Let the equation of the directrix be $y=mx+c$

So, the equation of the parabola : $$(x+1)^2+(y+1)^2=\dfrac{(mx-y+c)^2}{m^2+1}$$

First of all, it passes through $(7,13)$

Find the equation of the tangent at $(7,13)$ and compare with $3x–y–8=0$

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The tangent to a parabola at a point $P$ on it bisects the angle $\angle{FPD}$, where $F$ is the parabola’s focus and $D$ is the closest point on the directrix to $P$.

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Since $P$ is equidistant from $F$ and $D$, $D$ is the reflection of $F$ in the tangent line. By construction, the vector $P-D$ is perpendicular to the directrix, so an equation of the directrix is $(P-D)\cdot(X-D)=0$. You might need to divide the resulting equation by a common factor to get exactly the given solution.