Question

If in an ellipse, a focus is $(6,7)$, the corresponding directrix is $x+y+2=0$ and the eccentricity = $\frac{1}{2},$ then the equation of the ellipse is:

• A. $7x^2+2xy+7y^2-44x-108y+684=0$
• B. $7x^2-2xy+7y^2-100x-116y+676=0$
• C. $9x^2-2xy+9y^2-44x-108y+684$
• D. none of these

Answer 1

Answer: (B)

$7x^2-2xy+7y^2-100x-116y+676=0$

Ellipse is the locus of the point which moves in the plane such that its distance from a fixed point is a constant ratio from a fixed line(directrix). This fixed point is the locus.

$P$ is point on ellipse, $S$ is the focus and $PM$ is the distance of point $P$ from directrix.

Using definition of conic, $P{S}^2=e^2\cdot P{M}^2$

$\Rightarrow (x-6{)}^2+(y-7{)}^2=\frac{1}{4}{\left|\frac{x+y+2}{\sqrt{2}}\right|}^2$

$\Rightarrow 8(x^2+y^2-12x-14y+85)=x^2+y^2+2xy+4x+4y+4$

$\Rightarrow 7x^2-2xy+7y^2-100x-116y+676=0$