Question

Obtain the equation of the conic, a focus of which lies at (2,1), the directrix of

which is x+y = 0 and which passes through (1,4). Also identify the conic.

Answer 1

Given,

Focus of the conic is lies at $(2,1)$

The directrix of the conic lies at $,x+y=0$

It passes through the point $(1,4)$

Distance between the given points $(d)=\sqrt{1+9}=10$

The distance from the point $(1,4)$ from directrix $=\frac{5}{\sqrt{2}}$

The ratio of distance $=\frac{\sqrt{20}}{5}$

From the above, we can conclude that the ratio is less than 1, so it is an ellipse.

Hence the required equation is $\frac{(x-2)^2+(y-1)^2}{(\frac{(x+y)}{\sqrt{2}})^2}=(\frac{2}{\sqrt{5}})^2$

$\Rightarrow \frac{x^2+4-4x+y^2+1-2y}{\frac{x^2+y^2+2xy}{2}}=\frac{4}{5}$

$\Rightarrow 5x^2+20-20x+5y^2+5-10y=2x^2+2y^2+4xy$

$\Rightarrow 3x^2-4xy+3y^2-20x-10y+25=0$