Q. The normal at a point on the parabola $y^2 = 4x$ passes through $(5,0)$ If two more normals to this parabola also pass through $(5,0)$ then the centroid of the triangle formed by the feet of these three normals is
AP EAMCETAP EAMCET 2019
Solution:
Given equation of parabola $y^{2}=4 x$
Here, $a=1$
Equation of normal at point $P(5,0)$
$ y=-t x+2 t+t^{3} $
$ \Rightarrow 0=-5 t+2 t+t^{3} $
$ \Rightarrow 3 t=t^{3} $
$\Rightarrow t^{2}=3 \Rightarrow t=\pm \sqrt{3}, 0$
Co Normal Point
P
Q
R
$(at_1^2, 2at_1)$
$(at_2^2, 2at_2)$
$(at_3^2, 2at_3$
$= (3, 2\sqrt{3})$
$(3, -2\sqrt{3})$
$(0, 0)$
Hence, Centroid
$ = \left( \frac{3 + 3 + 0}{3}, \frac{2\sqrt{3} - 2\sqrt{3} + 0}{3} \right)$
$= \left(\frac{6}{3} , 0\right) = (2, 0)$
| Co Normal Point | P | Q | R |
|---|---|---|---|
| $(at_1^2, 2at_1)$ | $(at_2^2, 2at_2)$ | $(at_3^2, 2at_3$ | |
| $= (3, 2\sqrt{3})$ | $(3, -2\sqrt{3})$ | $(0, 0)$ |