The diagram shows an ellipse, centred at the origin with semi-minor axis $a$ and semi-major axis \

Question

The diagram shows an ellipse, centred at the origin with semi-minor axis $a$ and semi-major axis $2a$, where $a > 0$. The rectangle $PQRS$ is inscribed in the ellipse such that its four vertices $P$, $Q$, $R$ and $S$ lie on the ellipse. The coordinates of $P$ is $(x, y)$, where $x$ and $y$ are positive. Write down the cartesian equation of the ellipse in terms of $a$.[1] By considering the coordinates of $P$, show that $A$, the area of rectangle $PQRS$, is $8x\sqrt{a^2 - x^2}$.[2] Using differentiation, find, in terms of $a$, the value of $x$ when $A$ is a maximum. You do not need to prove that $A$ is a maximum.[3] Find the value of $a$ if the maximum value of $A$ is $100$.[2]

Accepted Answer

(i) $\frac{x^2}{a^2}+\frac{y^2}{4a^2}=1$ (ii) $A=8x\sqrt{a^2-x^2}$ $(\text{shown})$ (iii) $x=\frac{a}{\sqrt2}$ (iv) $a=5$

2024 JPJC Promo Q6

2024 JPJC Promo Q6

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