(A) The slope of the tangent $m = \tan(45^{\circ}) = 1$. For the parabola $y^2 = 4Ax$,the slope of the tangent at point $(At^2, 2At)$ is $m = 1/t$. Co

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$\left( \frac{a}{4}, \frac{a}{2} \right)$

(A) The slope of the tangent $m = \tan(45^{\circ}) = 1$. For the parabola $y^2 = 4Ax$,the slope of the tangent at point $(At^2, 2At)$ is $m = 1/t$. Comparing $y^2 = ax$ with $y^2 = 4Ax$,we get $4A = a$,so $A = a/4$. Since $m = 1/t = 1$,we have $t = 1$. The point of contact is $(At^2, 2At) = (\frac{a}{4}(1)^2, 2(\frac{a}{4})(1)) = (\frac{a}{4}, \frac{a}{2})$.

Class 11 Mathematics 10-2. Parabola, Ellipse, Hyperbola Parabola

Medium

If the tangent to the parabola $y^2 = ax$ makes an angle of $45^{\circ}$ with the $x$-axis,what is its point of contact?

A
$\left( \frac{a}{4}, \frac{a}{2} \right)$
B
$\left( \frac{a}{2}, \frac{a}{2} \right)$
C
$\left( -\frac{a}{2}, \frac{a}{4} \right)$
D
$\left( -\frac{a}{4}, \frac{a}{2} \right)$

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10-2. Parabola, Ellipse, Hyperbola

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If the tangent to the parabola $y^2 = ax$ makes an angle of $45^{\circ}$ with the $x$-axis,what is its point of contact?

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