An ellipse of equation $\frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{b}^{2}}}=1$, where \(0

Question

An ellipse of equation $\frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{b}^{2}}}=1$, where $0 < b < a$, has two points called foci ${{F}_{1}}\left( -c,0 \right)$ and ${{F}_{2}}\left( c,0 \right)$. The definition of the ellipse is such that for every point $P$ on the ellipse, the sum of the distance of $P$ to ${{F}_{1}}$ and ${{F}_{2}}$ is always a constant $k$. By considering one of the $x$-intercepts of the ellipse, determine the value of $k$ in terms of $a$ and/or $b$.[2] By considering another suitable point on the ellipse, find $c$ in terms of $a$ and $b$.[2] A hyperbola with equation ${{\left( y-h \right)}^{2}}-1=\frac{1}{4}{{\left( x-k \right)}^{2}}$ has $y=\frac{1}{2}x+\frac{3}{2}$ as one of its asymptotes, and the point $\left( 1,3 \right)$ is on the hyperbola. Find the values of $h$ and $k$.[4]

Accepted Answer

(a)(i) $k=2a$ (a)(ii) $c=\sqrt{a^2-b^2}$ (b) $h=2$, $k=1$

2021 HCI P2 Q2

2021 HCI P2 Q2

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