Answer to Question #200480 in Analytic Geometry for tanya

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

Answer:-

The distance of point (1,4)  from focus at (2,1) is

$\sqrt{ ( 1 − 2 )^2 + ( 4 − 1 )^2 } = \sqrt{ 10}$

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

and ratio of distances is $\frac{\sqrt{10}}{\frac{5}{\sqrt{2}}}=\frac{\sqrt{20}}{5}=\frac{2}{\sqrt{5}}<1$

As the ratio is less than  1, it is an ellipse

and the equation is obtained from ratio of the distance of a point on ellipse say (x,y) from focus (2,1) and its distance from directrix 

x+y=0 being $\frac{2}{\sqrt{5}}$ . the latter is $\frac{x+y}{\sqrt{2}}$ , hence equation is

$\frac{(x-2)^2+(y-1)^2}{(\frac{x+y}{\sqrt{2}})^2}$ =$(\frac{2}{\sqrt{5}})^2=\frac{4}{5}$

$\implies \boxed{2x^2-4xy+3y^2-20-10y+25=0}$