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Area between Two Curves

Suppose OABC ...

Question

Suppose $O A B C$ is a rectangle in the $x y -$ plane where $O$ is the origin and $A, B$ lie on the parabola $y = x^{2}$ . Then $C$ must lie on the curve

y = x^{2} + 2

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y = 2 x^{2} + 1

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y = x^{2} + 2

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y = - 2 x^{2} + 1

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Solution

The correct option is A $y = x^{2} + 2$

$∵ O B ⊥ O A ∴ m_{O B} . m_{O A} = t_{1} . t_{2} = - 1$
$t_{1} + t_{2} = h$ and $t_{1}^{2} + t_{2}^{2} = k$
$k = (t_{1} + t_{2})^{2} - 2 t_{1} . t_{2} = h^{2} + 2$
$∴ C must lie on the curve (y = x^{2} + 2)$

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