The locus of the orthocentre of the triangle | vedclass

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

Question

(A) Let the focal chord be $PQ$ where $P(at_1^2, 2at_1)$ and $Q(at_2^2, 2at_2)$. Since $PQ$ is a focal chord,$t_1t_2 = -1$.The normals at $P$ and $Q$

Vedclass · Accepted Answer

$y^2 = a(x - 3a)$

Vedclass · Answer

(A) Let the focal chord be $PQ$ where $P(at_1^2, 2at_1)$ and $Q(at_2^2, 2at_2)$. Since $PQ$ is a focal chord,$t_1t_2 = -1$.The normals at $P$ and $Q$ intersect at $R(h, k)$,where $h = a(t_1^2 + t_2^2 + t_1t_2 + 2)$ and $k = -at_1t_2(t_1 + t_2)$.Substituting $t_1t_2 = -1$ into the expressions:$h = a(t_1^2 + t_2^2 - 1 + 2) = a(t_1^2 + t_2^2 + 1)$$k = -a(-1)(t_1 + t_2) = a(t_1 + t_2)$Now,$(t_1 + t_2)^2 = t_1^2 + t_2^2 + 2t_1t_2 = t_1^2 + t_2^2 - 2$.So,$t_1^2 + t_2^2 = (t_1 + t_2)^2 + 2 = (k/a)^2 + 2$.Substituting this into the expression for $h$:$h = a((k/a)^2 + 2 + 1) = a(k^2/a^2 + 3) = k^2/a + 3a$.Rearranging for $k^2$:$k^2 = a(h - 3a)$.Replacing $(h, k)$ with $(x, y)$,the locus is $y^2 = a(x - 3a)$.

The locus of the orthocentre of the triangle formed by the focal chord of the parabola $y^2 = 4ax$ and the normals drawn at its extremities is

Similar Questions

The length of the latus rectum of a parabola,whose vertex and focus are on the positive $x$-axis at a distance $R$ and $S$ $(S > R)$ respectively from the origin,is:

Let $P(\alpha, \beta)$ be a point on the parabola $y^2 = 4x$. If $P$ also lies on the chord of the parabola $x^2 = 8y$ whose midpoint is $(1, 5/4)$,then $(\alpha - 28)(\beta - 8)$ is equal to:

The length of the latus rectum of the parabola $4y^2 + 2x - 20y + 17 = 0$ is

Let $(x, y)$ be any point on the parabola $y^2 = 4x$. Let $P$ be the point that divides the line segment from $(0, 0)$ to $(x, y)$ in the ratio $1:3$. Then the locus of $P$ is

Find the parametric coordinates of any point on the parabola whose focus is $(0, 1)$ and directrix is $x + 2 = 0$.

Vedclass Test Series

Exam Paper Generator

Online Exam Module