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Let 3x-y-8=0 be the equation of tangen...

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). Its directrix is

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We kown that
Length of latus rectum of parabola
`=2xx` Distance of focus from directrix
So, we need to find the equation of directrix.
Now, the image f focus in any tangent lies on the directrix Image of focus S(-1,-1) in the tangent y=3x-8 is the point N(5,-3), which lies on the directrix.
Also, line joining point of contact P(7, 13) and N(5, -3) is perpendicular to the directrix.
Slope of `NP=(13-(-3))/(7-5)=8`
`:.` Slope of directrix `=-(1)/(8)`
Since directrix passes through point N, its equation is x+8y+19=0.
So, length of latus rectum `=2xx(|-1+8(-1)+19|)/(sqrt(1+64))=(20)/(sqrt(65))`
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