From point P (6,0) three normals are drawn to the parabola y 2=8 x and points A , B and C are co-normal points. Then which of the following is/are correct?

Question

From point P (6,0) three normals are drawn to the parabola y 2=8 x and points A , B and C are co-normal points. Then which of the following is/are correct? (A) Area of triangle ABC is 8 (B) Circum-centre of triangle ABC is (5,0) (C) Centroid of triangle ABC is ((4/3), 0) (D)  triangle ABC is a right angled triangle

Tardigrade Admin · Accepted Answer

Let the normal be y=m x-4 m-2 m3 ⇒ 0=6 m -4 m -2 m 3 ⇒ m =0,1,-1 A (0,0) ; B (2,4) ; C (2,-4) text Area =8 text Centroid ≡((4/3), 0), text circumcentre ≡(5,0)

Q. From point $P (6, 0)$ three normals are drawn to the parabola $y^{2} = 8 x$ and points $A, B$ and $C$ are co-normal points. Then which of the following is/are correct?

Area of $△ A BC$ is 8

Circum-centre of $△ A BC$ is $(5, 0)$

Centroid of $△ A BC$ is $(\frac{4}{3}, 0)$

$△ A BC$ is a right angled triangle

Let the normal be $y = m x - 4 m - 2 m^{3}$
$\Rightarrow 0 = 6 m - 4 m - 2 m^{3} \Rightarrow m = 0, 1, - 1$
$A (0, 0); B (2, 4); C (2, - 4)$
$Area = 8$
$Centroid \equiv (\frac{4}{3}, 0), circumcentre \equiv (5, 0)$

Q. From point P(6,0) three normals are drawn to the parabola y2=8x and points A,B and C are co-normal points. Then which of the following is/are correct?

Area of △ABC is 8

Circum-centre of △ABC is (5,0)

Centroid of △ABC is (34​,0)

△ABC is a right angled triangle