Let y=3x-8 be the equation of the tangent at the point (7, 13 ) lying on a parabola whose focus is at (-1,-1). Find the equation of directrix and the length of the latus rectum of the parabola.

Explanation

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))`