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Byju's Answer

Standard XII

Mathematics

Definition of Ellipse

Let C1 ...

Question

Let $C_{1}$ & $C_{2}$ be two curves passing through the origin as shown in the figure A curve C is said to "bisect the area" the region between $C_{1}$ & $C_{1}$ if for each point P of C the two shaded regions A & B shown in the figure have equal areas Determine the upper curve $C_{2}$ given that the bisecting curve C has the equation $y = x^{2}$ & that the lower curve $C_{1}$ has the equation $y = x^{2} / 2$

(16 / 9) x^{2}

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(25 / 9) x^{2}

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(25 / 16) x^{2}

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(9 / 25) x^{2}

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Solution

The correct option is A $(16 / 9) x^{2}$
According to question
$\int_{0}^{a^{2}} (- f^{- 1} (y) + \sqrt{y}) d y = \int_{0}^{a} (x^{2} - \frac{x^{2}}{2}) d x$
$\Rightarrow [f^{- 1} (a^{2}) - a] 2 a = - \frac{a^{2}}{2} \Rightarrow f^{- 1} (a^{2}) = \frac{3 a}{4} \Rightarrow f (\frac{3 a}{4}) - a^{2}$
or $f (x) = \frac{16}{9} x^{2}$

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