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

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(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},

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

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 $Q$.

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