If the normals at two points on the parabola | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(B) Let the two points on the parabola $y^2 = 4ax$ be $P(at_1^2, 2at_1)$ and $Q(at_2^2, 2at_2)$.The equation of the normal at point $t$ is $y = -tx +

Vedclass · Accepted Answer

$8a^2$

Vedclass · Answer

(B) Let the two points on the parabola $y^2 = 4ax$ be $P(at_1^2, 2at_1)$ and $Q(at_2^2, 2at_2)$.The equation of the normal at point $t$ is $y = -tx + 2at + at^3$.If the normals at $t_1$ and $t_2$ meet at a point $(x, y)$ on the parabola,then the point of intersection of the normals is given by $x = 2a + a(t_1^2 + t_1t_2 + t_2^2)$ and $y = -at_1t_2(t_1 + t_2)$.Since the intersection point lies on the parabola $y^2 = 4ax$,we have $(-at_1t_2(t_1 + t_2))^2 = 4a(2a + a(t_1^2 + t_1t_2 + t_2^2))$.This simplifies to $a^2t_1^2t_2^2(t_1 + t_2)^2 = 4a^2(2 + t_1^2 + t_1t_2 + t_2^2)$.For the normals to meet on the parabola,the condition is $t_1t_2 = 2$.The ordinates of the points are $y_1 = 2at_1$ and $y_2 = 2at_2$.The product of the ordinates is $y_1y_2 = (2at_1)(2at_2) = 4a^2(t_1t_2)$.Substituting $t_1t_2 = 2$,we get $y_1y_2 = 4a^2(2) = 8a^2$.

If the normals at two points on the parabola $y^2 = 4ax$ meet on the parabola,then what is the product of the ordinates of these two points?

Similar Questions

Let $A$ be the focus of the parabola $y^{2}=8x$. Let the line $y=mx+c$ intersect the parabola at two distinct points $B$ and $C$. If the centroid of the triangle $ABC$ is $(\frac{7}{3},\frac{4}{3})$, then $(BC)^{2}$ is equal to:

The tangents drawn from the ends of the latus rectum of the parabola ${y^2} = 12x$ meet at:

If the line $y = 2x + k$ is a tangent to the curve $x^2 = 4y$,then $k$ is equal to

In which quadrant does the vertex of the parabola $y^2 + 2y + x = 0$ lie?

Let the image of the parabola $x^{2} = 4y$ in the line $x - y = 1$ be $(y + a)^{2} = b(x - c)$, where $a, b, c \in \mathbb{N}$. Then $a + b + c$ is equal to

Vedclass Test Series

Exam Paper Generator

Online Exam Module