If the tangent to the parabola y^{2} = 4ax at | vedclass

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

Question

(A) Let the points $P$ and $Q$ on the parabola $y^{2} = 4ax$ be represented by parameters $t_{1}$ and $t_{2}$ respectively,such that $P = (at_{1}^{2},

Vedclass · Accepted Answer

$(a^{2}/p, -4a^{2}/q)$

Vedclass · Answer

(A) Let the points $P$ and $Q$ on the parabola $y^{2} = 4ax$ be represented by parameters $t_{1}$ and $t_{2}$ respectively,such that $P = (at_{1}^{2}, 2at_{1})$ and $Q = (at_{2}^{2}, 2at_{2})$.The slope of the tangent at any point $t$ is $m = 1/t$.Since the tangents at $P$ and $Q$ are perpendicular,the product of their slopes is $-1$:$(1/t_{1}) 	imes (1/t_{2}) = -1 \implies t_{1}t_{2} = -1 \implies t_{2} = -1/t_{1}$.Given $P(p, q) = (at_{1}^{2}, 2at_{1})$,we have $q = 2at_{1} \implies t_{1} = q/(2a)$.Thus,$t_{2} = -1/(q/2a) = -2a/q$.The coordinates of $Q$ are $(at_{2}^{2}, 2at_{2}) = (a(-2a/q)^{2}, 2a(-2a/q)) = (a(4a^{2}/q^{2}), -4a^{2}/q)$.Since $P(p, q)$ lies on $y^{2} = 4ax$,we have $q^{2} = 4ap$. Substituting $q^{2} = 4ap$ into the $x$-coordinate of $Q$:$x_{Q} = (4a^{3})/(4ap) = a^{2}/p$.Therefore,the coordinates of $Q$ are $(a^{2}/p, -4a^{2}/q)$.

If the tangent to the parabola $y^{2} = 4ax$ at point $P(p, q)$ is perpendicular to the tangent at another point $Q$,find the coordinates of $Q$.

Similar Questions

The length of the latus rectum of the conic $25[(x-2)^2+(y-3)^2]=(3x-4y+7)^2$ is

The equation of the tangent to the parabola $y^2=12x$ at the point $(3,-6)$ is:

Let a focal chord $12x + 5y - 27 = 0$ of the parabola $y^2 = kx$ intersect the parabola at the points $P$ and $P^{\prime}$. If $S$ is the focus of this parabola,then $9(SP + SP^{\prime}) = $

If $2x + 3y + 12 = 0$ and $x - y + 4\lambda = 0$ are conjugate with respect to the parabola $y^2 = 8x$,then $\lambda$ is equal to

The point $(3,4)$ is the focus and $2x - 3y + 5 = 0$ is the directrix of a parabola. Its latus rectum is:

Vedclass Test Series

Exam Paper Generator

Online Exam Module