Question

Find the equation of the ellipse in the following cases:
(i) focus is (0, 1), directrix is x + y = 0 and e = $\frac{1}{2}$
(ii) focus is (−1, 1), directrix is x − y + 3 = 0 and e = $\frac{1}{2}$
(iii) focus is (−2, 3), directrix is 2x + 3y + 4 = 0 and e = $\frac{4}{5}$
(iv) focus is (1, 2), directrix is 3x + 4y − 5 = 0 and e = $\frac{1}{2}$.

Answer 1

(i)



$$\begin{gathered} \mathrm{Let}S(0,1)\mathrm{be}\mathrm{the}\mathrm{focus}\mathrm{and}\mathrm{ZZ}\text{'}\mathrm{be}\mathrm{the}\mathrm{directrix}. \\ \mathrm{Let}\mathrm{P}(x,y)\mathrm{be}\mathrm{any}\mathrm{point}\mathrm{on}\mathrm{the}\mathrm{ellipse}\mathrm{and}\mathrm{let}\mathrm{PM}\mathrm{be}\mathrm{the}\mathrm{perpendicular}\mathrm{from}\mathrm{P}\mathrm{on}\mathrm{the}\mathrm{directrix}. \\ \mathrm{Then}\mathrm{by}\mathrm{the}\mathrm{definition},\mathrm{we}\mathrm{have}: \\ \mathrm{SP}=e\times \mathrm{PM} \\ \Rightarrow \mathrm{SP}=\frac{1}{2}\times \mathrm{PM} \\ \Rightarrow 2\mathrm{SP}=\mathrm{PM} \\ \Rightarrow 4{\left(\mathrm{SP}\right)}^2={\mathrm{PM}}^2 \\ \Rightarrow 4\left[{\left(x\right)}^2+{\left(y-1\right)}^2\right]={\left|\frac{x+y}{\sqrt{1^2+{\left(1\right)}^2}}\right|}^2 \\ \Rightarrow 4\left[x^2+y^2+1-2y\right]=\frac{x^2+y^2+2xy}{2} \\ \Rightarrow 8x^2+8y^2+8-16y=x^2+y^2+2xy \\ \Rightarrow 7x^2+7y^2-2xy-16y+8=0 \\ \text{This is the required equation of the ellipse.} \end{gathered}$$

(ii)

$$\begin{gathered} \mathrm{Let}S(-1,1)\mathrm{be}\mathrm{the}\mathrm{focus}\mathrm{and}\mathrm{ZZ}\text{'}\mathrm{be}\mathrm{the}\mathrm{directrix}. \\ \mathrm{Let}\mathrm{P}(x,y)\mathrm{be}\mathrm{any}\mathrm{point}\mathrm{on}\mathrm{the}\mathrm{ellipse}\mathrm{and}\mathrm{let}\mathrm{PM}\mathrm{be}\mathrm{the}\mathrm{perpendicular}\mathrm{from}\mathrm{P}\mathrm{on}\mathrm{the}\mathrm{directrix}. \\ \mathrm{Then}\mathrm{by}\mathrm{the}\mathrm{definition},\mathrm{we}\mathrm{have}: \\ \mathrm{SP}=e\times \mathrm{PM} \\ \Rightarrow \mathrm{SP}=\frac{1}{2}\times \mathrm{PM} \\ \Rightarrow 2\mathrm{SP}=\mathrm{PM} \\ \Rightarrow 4{\left(\mathrm{SP}\right)}^2={\mathrm{PM}}^2 \\ \Rightarrow 4\left[{\left(x+1\right)}^2+{\left(y-1\right)}^2\right]={\left|\frac{x-y+3}{\sqrt{1^2+{\left(-1\right)}^2}}\right|}^2 \\ \Rightarrow 4\left[x^2+1+2x+y^2+1-2y\right]=\frac{x^2+y^2+9-2xy-6y+6x}{2} \\ \Rightarrow 8x^2+8+16x+8y^2+8-16y=x^2+y^2+9-2xy-6y+6x \\ \Rightarrow 7x^2+7y^2+2xy-10y+10x+7=0 \\ \text{This is the required equation of the ellipse.} \end{gathered}$$

(iii)


$$\begin{gathered} \mathrm{Let}S(-2,3)\mathrm{be}\mathrm{the}\mathrm{focus}\mathrm{and}\mathrm{ZZ}\text{'}\mathrm{be}\mathrm{the}\mathrm{directrix}. \\ \mathrm{Let}\mathrm{P}(x,y)\mathrm{be}\mathrm{any}\mathrm{point}\mathrm{on}\mathrm{the}\mathrm{ellipse}\mathrm{and}\mathrm{let}\mathrm{PM}\mathrm{be}\mathrm{the}\mathrm{perpendicular}\mathrm{from}\mathrm{P}\mathrm{on}\mathrm{the}\mathrm{directrix}. \\ \mathrm{Then}\mathrm{by}\mathrm{the}\mathrm{definition},\mathrm{we}\mathrm{have}: \\ \mathrm{SP}=e\times \mathrm{PM} \\ \Rightarrow \mathrm{SP}=\frac{4}{5}\times \mathrm{PM} \\ \Rightarrow \frac{5}{4}SP=\mathrm{PM} \\ \Rightarrow \frac{25}{16}{\left(\mathrm{SP}\right)}^2={\mathrm{PM}}^2 \\ \Rightarrow \frac{25}{16}\left[{\left(x+2\right)}^2+{\left(y-3\right)}^2\right]={\left|\frac{2x+3y+4}{\sqrt{2^2+3^2}}\right|}^2 \\ \Rightarrow \frac{25}{16}\left[x^2+4+4x+y^2+9-6y\right]=\frac{4x^2+9y^2+16+12xy+24y+16x}{13} \\ \Rightarrow 325\left(x^2+4+4x+y^2+9-6y\right)=16\left(4x^2+9y^2+16+12xy+24y+16x\right) \\ \Rightarrow 325x^2+1300+1300x+325y^2+2925-1950y=64x^2+144y^2+256+192xy+384y+256x \\ \Rightarrow 261x^2+181y^2+1044x-2309y-192xy+3969=0 \\ \text{This is the required equation of the ellipse.} \end{gathered}$$

(iv)

$$\begin{gathered} \mathrm{Let}\mathrm{S}(1,2)\mathrm{be}\mathrm{the}\mathrm{focus}\mathrm{and}\mathrm{ZZ}\text{'}\mathrm{be}\mathrm{the}\mathrm{directrix}. \\ \mathrm{Let}\mathrm{P}(x,y)\mathrm{be}\mathrm{any}\mathrm{point}\mathrm{on}\mathrm{the}\mathrm{ellipse}\mathrm{and}\mathrm{let}\mathrm{PM}\mathrm{be}\mathrm{the}\mathrm{perpendicular}\mathrm{from}\mathrm{P}\mathrm{on}\mathrm{the}\mathrm{directrix}. \\ \mathrm{Then}\mathrm{by}\mathrm{the}\mathrm{definition},\mathrm{we}\mathrm{have}: \\ \mathrm{SP}=e\times \mathrm{PM} \\ \Rightarrow \mathrm{SP}=\frac{1}{2}\times \mathrm{PM} \\ \Rightarrow 2\mathrm{SP}=\mathrm{PM} \\ \Rightarrow 4{\left(\mathrm{SP}\right)}^2={\mathrm{PM}}^2 \\ \Rightarrow 4\left[{\left(x-1\right)}^2+{\left(y-2\right)}^2\right]={\left|\frac{3x+4y-5}{\sqrt{3^2+{\left(4\right)}^2}}\right|}^2 \\ \Rightarrow 4\left[x^2+1-2x+y^2+4-4y\right]=\frac{9x^2+16y^2+25+24xy-40y+30x}{25} \\ \Rightarrow 100x^2+100-200x+100y^2+400-400y=9x^2+16y^2+25+24xy-40y-30x \\ \Rightarrow 91x^2+84y^2-24xy-360y-170x+475=0 \\ \text{This is the required equation of the ellipse.} \end{gathered}$$