Question

If the equation of the tangent at the point $P\left(3,4\right)$ on the parabola whose axis is the $x$ -axis is $3x-4y+7=0,$ then the distance of the tangent from the focus of the parabola is

Answer 1

Slope of the tangent is $\frac{3}{4}=tan \theta $
Slope of the focal chord through $P\left(3,4\right)$ is $tan 2\theta =\frac{2 tan ⁡ \theta }{1 - t a n^{2} \theta }$

i.e. $tan2\theta =\frac{2 \left(\frac{3}{4}\right)}{1 - \frac{9}{16}}=\frac{24}{7}$
Equation of the focal chord through $P$ is $\left(y - 4\right)=\frac{24}{7}\left(x - 3\right)$
Focus is $S\left(\frac{11}{6} , 0\right)$
Distance of the focus from the tangent is
$\frac{\left|3 \left(\frac{11}{6}\right) - 4 \left(0\right) + 7\right|}{\sqrt{3^{2} + 4^{2}}}=\frac{5}{2}=2.5$