Question

Determine the coordinates of the point where tangents drawn at
the points $(1,2)$ and $(4,4)$ to the parabola $y^2=4x$ meet.

Answer 1

both the points $(1,2)$ and $(4,4)$ lie on the parabola : $y^2=4x$

$\because$ Tangent to the parabola from a point on it is given by equation : $y{y}_1=2(x+x_1)$

$\therefore$ tangent to the parabola from point $(1,2)$ is $2y=2(x+1)$ .....(i)

Tangent to the parabola from point $(4,4)$ is $4y=2(x+4)$.....(ii)

(i)$\times 2-$(ii) we get,

$\Rightarrow 4y-4y=4(x+1)-2(x+4)$

$\Rightarrow 4x+4-2x-8=0$

$\Rightarrow 2x=4$

$\Rightarrow x=2$

$y=(2+1)=3$

$\therefore$ both tangents intersect at $(2,3)$