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Q.
The combined equation of directrices of the ellipse whose eccentricity is $\frac{1}{2}$, one of the foci is $(1,1)$ and minor axis is $x+y=4$ is
Conic Sections
Solution:
Equation of major axis: $x - y =\lambda$, passes through $(1,1) \Rightarrow \lambda=0$ $3 /$ ellisC $\therefore x - y =0$
Hence, centre is $(2,2)$
Now, $CS =$ ae
$\Rightarrow \sqrt{2}= a \cdot \frac{1}{2} \Rightarrow a =2 \sqrt{2} \text {. }$
Equation of directrix: $x+y=\lambda$
$\text { perpendicular from centre to directrix }=\frac{ a }{ e } $
$\Rightarrow\left|\frac{\lambda-4}{\sqrt{2}}\right|=\frac{2 \sqrt{2}}{1 / 2}=4 \sqrt{2} $
$\lambda-4= \pm 8 $
$\lambda=12 \text { or }-4$
$\therefore \text { equation of directrix are } x+y-12=0 \text { and } x+y+4=0$
$\therefore \text { combined equation: }(x+y-12)(x+y+4)=0$
$(x+y)^2-8(x+y)-48=0 $
