Question

find the area bounded by y2=4ax and the tangents at the ends of latus rectum.

Answer 1

Hey friend, Harish here.

Here is your answer.

The equation of parabola given is y² = 4ax . ( Represented by the red line in the image.)

And the Latus rectum is a line with equation x = a. (Blue line).

Area Bounded by the line and the Parabola = 2 × Area of the first quadrant.

$\mathrm{Area= 2 \int \limits_0^a y\ dx} \\ \\ \implies \mathrm{2 \int \limits _0^a 2 \sqrt {a} .x^{\large\frac{1}{2}}\ dx}} \\ \\ \implies 4 \sqrt a \bigg [\large\frac{x}{3}}.\ x^{\large\frac {3}{2}}\bigg]_0^a \\ \\ \implies \frac{8}{3} . a^{\frac{1}{2}} a^{\frac{3}{2}} \\ \\ \boxed{\mathrm{Area = \frac{8}{3}}a^2}$

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Hope my answer is helpful to you.