Conic portions from equations with a square root

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Sketch the following conic portions after setting the conditions on $x$ and $y$; then find semi-axes, vertices, foci and (if a hyperbola) the asymptotes.

• (a) $y=2+3\sqrt{4-x^2}$
• (b) $y=-1+2\sqrt{x^2-4x-5}$
• (c) $y=3-2\sqrt{x^2+6x+13}$

Solution

(a) DC $-2\le x\le2$; also $y\ge2$. Squaring: $\dfrac{x^2}{4}+\dfrac{(y-2)^2}{36}=1$: ellipse centre $(0;2)$, $a=2$ (horiz.), $b=6$ (vert.), $c=4\sqrt2$, foci $(0;2\pm4\sqrt2)$. Portion: the upper half ($y\ge2$).

(b) DC $x\le-1\vee x\ge5$; $y\ge-1$. Squaring: $\dfrac{(x-2)^2}{9}-\dfrac{(y+1)^2}{36}=1$: hyperbola, horizontal, centre $(2;-1)$, $a=3$, $b=6$, vertices $(-1;-1),(5;-1)$, asymptotes $y+1=\pm2(x-2)$. Portion: the upper halves of both branches.

(c) DC $\forall x$; $y\le3$. Squaring: $\dfrac{(y-3)^2}{16}-\dfrac{(x+3)^2}{4}=1$: hyperbola, vertical, centre $(-3;3)$, $a=4$ (vert.), $b=2$, vertices $(-3;7),(-3;-1)$, asymptotes $y-3=\pm2(x+3)$. Portion: the lower halves ($y\le3$).

$\boxed{\text{(a) ellipse (upper arc); (b),(c) hyperbolas (half branches)}}$