Question #200480

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.


1
Expert's answer
2021-06-01T18:18:47-0400

Answer:-

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

(12)2+(41)2=10\sqrt{ ( 1 − 2 )^2 + ( 4 − 1 )^2 } = \sqrt{ 10}

The distance of point(1,4) from directrix is 1+42=52\frac{1+4}{\sqrt{2}}=\frac{5}{\sqrt{2}}

and ratio of distances is 1052=205=25<1\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 25\frac{2}{\sqrt{5}} . the latter is x+y2\frac{x+y}{\sqrt{2}} , hence equation is

(x2)2+(y1)2(x+y2)2\frac{(x-2)^2+(y-1)^2}{(\frac{x+y}{\sqrt{2}})^2} =(25)2=45(\frac{2}{\sqrt{5}})^2=\frac{4}{5}

    2x24xy+3y22010y+25=0\implies \boxed{2x^2-4xy+3y^2-20-10y+25=0}




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