The normal at a point on the parabola y2 = 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

Question

The normal at a point on the parabola y2 = 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 (A) ( (1/2) , (1/2) ) (B) ( 2 , 0 ) (C) ( 5 , 0 ) (D) ( 0 , 2)

Tardigrade Admin · Accepted Answer

Given equation of parabola y2=4 x Here, a=1 Equation of normal at point P(5,0) y=-t x+2 t+t3 ⇒ 0=-5 t+2 t+t3 ⇒ 3 t=t3 ⇒ t2=3 ⇒ t=± √3, 0 Co Normal Point P Q R (at12, 2at1) (at22, 2at2) (at32, 2at3 = (3, 2√3) (3, -2√3) (0, 0) Hence, Centroid = ( (3 + 3 + 0/3), (2√3 - 2√3 + 0/3) ) = ((6/3) , 0) = (2, 0)

Co Normal Point	P	Q	R
	$(a t_{1}^{2}, 2 a t_{1})$	$(a t_{2}^{2}, 2 a t_{2})$	$(a t_{3}^{2}, 2 a t_{3}$
	$= (3, 23)$	$(3, - 23)$	$(0, 0)$

Q. The normal at a point on the parabola $y^{2} = 4 x$ 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

$(\frac{1}{2}, \frac{1}{2})$

$(2, 0)$

$(5, 0)$

$(0, 2)$

Q. The normal at a point on the parabola y2=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

(21​,21​)

(2,0)

(5,0)

(0,2)

Q. The normal at a point on the parabola $y^{2} = 4 x$ 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

$(\frac{1}{2}, \frac{1}{2})$

$(2, 0)$

$(5, 0)$

$(0, 2)$