Question

Write down the parametric equations of the following ellipses. Find their eccentricities, foci and directrices. (a) 4x² + 9y2 = 4 (b) x²/9 + y²/16 = 1 (c) x²/7 + y²/14 = 1

Answer 1

(a)

$$4x^2+9y^2=4$$

$$\dfrac{x^2}{1}+\dfrac{y^2}{4/9}=1$$

$$x=\cos t, y=\dfrac{2}{3}\sin t, 0\leq t\leq 2\pi$$

$$\vec r=\langle\cos t, \dfrac{2}{3}\sin t\rangle, 0\leq t\leq 2\pi$$

$$a=1, b=\dfrac{2}{3}, c=\sqrt{a^2-b^2}=\sqrt{(1)^2-(\dfrac{2}{3})^2}=\dfrac{\sqrt{5}}{3}$$

$$e=\dfrac{c}{a}=\dfrac{\sqrt{5}}{3}$$

$$Foci: (-\dfrac{\sqrt{5}}{3}, 0), (\dfrac{\sqrt{5}}{3}, 0)$$

$$Directrices: x=\pm\dfrac{a^2}{c}$$

$$first\ directrix: x=-\dfrac{3\sqrt{5}}{5},$$

$$second\ directrix:x=\dfrac{3\sqrt{5}}{5}$$

(b)

$$\dfrac{x^2}{9}+\dfrac{y^2}{16}=1$$

$$x=3\cos t, y=4\sin t, 0\leq t\leq 2\pi$$

$$\vec r=\langle3\cos t, 4\sin t\rangle, 0\leq t\leq 2\pi$$

$$a=4, b=3, c=\sqrt{a^2-b^2}=\sqrt{(4)^2-(3)^2}=\sqrt{7}$$

$$e=\dfrac{c}{a}=\dfrac{\sqrt{7}}{4}$$

$$Foci: (-\sqrt{7}, 0), (\sqrt{7}, 0)$$

$$Directrices: y=\pm\dfrac{a^2}{c}$$

$$first\ directrix: y=-\dfrac{16\sqrt{7}}{7},$$

$$second\ directrix:y=\dfrac{16\sqrt{7}}{7}$$

(c)

$$\dfrac{x^2}{7}+\dfrac{y^2}{14}=1$$

$$x=\sqrt{7}\cos t, y=\sqrt{14}\sin t, 0\leq t\leq 2\pi$$

$$\vec r=\langle\sqrt{7}\cos t, \sqrt{14}\sin t\rangle, 0\leq t\leq 2\pi$$

$$a=\sqrt{14}, b=\sqrt{7}, c=\sqrt{a^2-b^2}=\sqrt{(\sqrt{14})^2-(\sqrt{7})^2}=\sqrt{7}$$ $$e=\dfrac{c}{a}=\dfrac{\sqrt{2}}{2}$$

$$Foci: (-\sqrt{7}, 0), (\sqrt{7}, 0)$$

$$Directrices: y=\pm\dfrac{a^2}{c}$$

$$first\ directrix: y=-2\sqrt{7},$$

$$second\ directrix:y=2\sqrt{7}$$