Let ABCD be a trapezium whose vertices lie on the parabola . Let the sides AD and BC of the trapezium be parallel to y-axis. If the diagonal AC is of length and it passes through the point (1, 0), then the area of ABCD is [2025]

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Solution

Q.
Let ABCD be a trapezium whose vertices lie on the parabola $y^{2} = 4 x$ . Let the sides AD and BC of the trapezium be parallel to y-axis. If the diagonal AC is of length $\frac{25}{4}$ and it passes through the point (1, 0), then the area of ABCD is [2025]

1 $\frac{25}{2}$

2 $\frac{125}{8}$

3 $\frac{75}{4}$

4 $\frac{75}{8}$

Ans.
(3)

Let $A (a t_{1}^{2}, 2 a t_{1})$ and $C (\frac{a}{t_{1}^{2}}, \frac{- 2 a}{t_{1}})$ be the points lies on parabola $y^{2} = 4 x$ .

$∴$ Length of $A C = a {(t_{1} + \frac{1}{t_{1}})}^{2} = \frac{25}{4}$ (Given)

$\Rightarrow t_{1} + \frac{1}{t_{1}} = \pm \frac{5}{2}$

So, $A (\frac{1}{4}, 1), B (4, 4), C (4, - 4)$ and $D (\frac{1}{4}, - 1)$

$∴$ The area of trapezium ABCD = $\frac{1}{2} (8 + 2) (4 - \frac{1}{4}) = \frac{75}{4}$ .

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