# Find the area of triangle formed by the tangents of parabola and chord of contacts

Find the area of the triangle formed by the tangents from the point $(h, k)$ to the parabola $y^2=4ax$ and the chord of contact.

I find the the tangents from external points $(h,k)$, i.e. $$(y^2-4ax)(k^2-4ah)=(ky-2a(x+h))^2$$ Again the chord of contact of the given parabola is $ky=2a(x+h)$. Then we try to find out three extremities.

• I find the the tangents from external points $(h,K)$, i.e. $(y^2-4ax)(k^2-4ah)=(ky-2a(x+h))$. again the chord of contact of the given parabola is $ky=2a(x+h)$. Then we try to find out three extremites
– MSMM
May 25 '17 at 3:11

Let $$(x_1,y_1)$$ and $$(x_2,y_2)$$ be the points of contact.
Substitute the chord $$ky=2a(x+h)$$ into the parabola $$y^2=4ax$$ to obtain a quadratic equation in $$y$$. $$y_1$$ and $$y_2$$ are the roots. So we have $$y_1+y_2$$ and $$y_1y_2$$. $$(y_1-y_2)^2=(y_1+y_2)^2-4y_1y_2$$.
$$k(y_1-y_2)=2a(x_1-x_2)$$ and so we can find the distance between the two points of contact.
The distance between the point $$(h,k)$$ and the chord of contact is $$\displaystyle \left|\frac{2a(h)-k(k)+2ah}{\sqrt{4a^2+k^2}}\right|$$.
Multiply the two distances and divide it by $$2$$, we have the area.