Finding the length of the chord of contact of the parabola.

Question

Tangents are drawn to the parabola $y^2= 4x$ from the point $(1,3)$ What is the length of the chord of contact?

The chord of contact is $3y-2x-2=0$

The points of intersection of this chord with the parabola are : $(\dfrac{7 \pm \sqrt{45}}{2},\dfrac{6 \pm \sqrt{20}}{2})$

Using distance formula I get the length to be $\dfrac{\sqrt{65}}{2}$ but answer given is $\sqrt{65}.$ Where have I gone wrong?

Answer 1

You have the correct coordinates, but if you check the distance formula the answer is indeed $\sqrt{65}$ not $\frac 12\sqrt{65}$

Answer 2

The length of any chord in a parabola is given as:

$$\frac{4}{m^2}\sqrt{a(1+m^2)(a-mc)}$$

where m is the slope of the chord whose length is to be calculated and $c$ is the constant $t$ of that chord.