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) Let P(x,y) be a point on the ellipse.
Then, by definition $SP=ePM$
Here $e=\frac{1}{2}$ coordinates of S are (0,1) and the equation of the directrix is $x+y=0$.
$\therefore SP=\frac{1}{2}\left(PM\right)$
$\Rightarrow S{P}^2=\frac{1}{4}(PM{)}^2$
$\Rightarrow 4S{P}^2=(PM{)}^2$
$\Rightarrow 4\left[\right(x-0{)}^2+(y-1{)}^2]=\left[\frac{x+y}{\sqrt{1^2+1^2}}\right]$
$\Rightarrow 4[x^2+y^2+1-2y]=\frac{(x+y{)}^2}{2}$
$\Rightarrow 4\times 2[x^2+y^2-2y+1]=x^2+y^2+2xy$
$\Rightarrow 8x^2+8y^2-16y+8=x^2+y^2+2xy$
$\Rightarrow 8x^2-x^2+8y^2-y^2-2xy-16y+8=0$
$\Rightarrow 7x^2+7y^2-2xy-16y+8=0$

(ii)Let P(x,y) be a point on the ellipse. Then, by definition
SP= ePM
Here $e=\frac{1}{2}$ coordinates of S are (-1,1) and the equation of directrix is $x-y+3=0$
$\therefore SP=\frac{1}{2}PM$
$\Rightarrow S{P}^2=\frac{1}{4}(PM{)}^2$
$\Rightarrow 4S{P}^2=(PM{)}^2$
$\Rightarrow 4\left[\right(x+1{)}^2+(y-1{)}^2]={\left[\frac{x-y+3}{\sqrt{1^2+(-1{)}^2}}\right]}^2$
$\Rightarrow 4[x^2+1+2x+y^2+1-2y]=\frac{(x-y+3{)}^2}{2}$
$\Rightarrow 8[x^2+y^2+2x-2y+2]=(x-y+3{)}^2$
$\Rightarrow 8x^2+8y^2+16x-16y+16=x^2+(-y{)}^2+3^2+2\times x\times (-y)+2\times 3\times x+2\times 3\times (-y)$
$\Rightarrow 8x^2+8y^2+16x-16y+16=x^2+y^2+9-6y-2xy+6x$
$\Rightarrow 8x^2-x^2+8y^2-y^2+2xy+16x-6x-16y+16+6y-9=0$
$\Rightarrow 7x^2+7y^2+2xy+10x-10y+7=0$
(iii) Let P(x,y) be a point on the ellipse. Then by definition
SP=e PM
Here $e=\frac{4}{5},$ coordinates of S are (-2,3) and the equation of directrix is 2x+3y+4=0
$\therefore SP=\frac{4}{5}PM$