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

Answer: 2.5

Slope of the tangent is $\frac{3}{4}=tan\theta$
Slope of the focal chord through $P\left(3,4\right)$ is $tan2\theta =\frac{2tan\theta }{1-ta{n}^2\theta }$

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
$\frac{|3\left(\frac{11}{6}\right)-4\left(0\right)+7|}{\sqrt{3^2+4^2}}=\frac{5}{2}=2.5$