Question

If the focus and corresponding directrix of an ellipse are $(3,4)$ and $x+y-1=0$ respectively and ecentricity is $\frac{1}{2}$ then find the coordinates of extremities of major axis.

Answer 1

REF.Image.

Given focus =(3,4)

directrix $=x+y-1=0$

eccentricity (e) = $\frac{1}{2}$

Equation of axis $(y-y_1)=x-x_1$

$y-4=x-3$

$y=x+1$

distance from (3,4) $x_1,x_2$ to $x+y-1=0$

$a=1,b=1,c=-1$

$d=\frac{a{x}_1+b{y}_1+c}{\sqrt{a^2+b^2}}$

$d=\frac{1\left(3\right)+1\left(4\right)-1}{\sqrt{1+1}}=\frac{3+4-1}{\sqrt{2}}=\frac{6}{\sqrt{2}}$

$d=\frac{a}{e}-ae$

So $\frac{a}{e}-ae=\frac{6}{\sqrt{2}}[\because e=\frac{1}{2}]$

$2a-\frac{a}{2}=\frac{6}{\sqrt{2}}\Rightarrow 4a-a=\frac{12}{\sqrt{2}}\Rightarrow 3a=\frac{6\times 2}{\sqrt{2}}$

$\therefore a=2\sqrt{2}$ major axis