Question

Let A, B and C be the vertices of a variable right angled triangle inscribed in the parabola $y^2 = 16x$. Let the vertex B containing the right angle be $(4, 8)$ and the locus of the centroid of $\Delta ABC$ be a conic $C_0$. Then three times the length of latus rectum of $C_0$ is _______.

Answer 1

Correct Answer: 16

For parabola \[ y^2=16x \] we have \[ a=4 \] Parametric point is \[ (4t^2,8t) \] Given $B=(4,8)$ corresponds to \[ t=1 \] Let other points be parameters $t_1,t_2$. Condition of right angle at $B$: \[ m_1m_2=-1 \] Using slope of chord formula \[ \frac{2}{1+t_1}\cdot \frac{2}{1+t_2}=-1 \] \[ (1+t_1)(1+t_2)=-4 \] After centroid calculation and simplifying, locus becomes \[ y^2=\frac{16}{3}\left(x-\frac{40}{3}\right) \] For parabola \[ y^2=4ax \] length of latus rectum is \[ 4a \] So here \[ L.R.=\frac{16}{3} \] Required: \[ 3\times \frac{16}{3}=16 \] Final Answer: \[ \boxed{16} \]