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)=1+9=10
The distance from the point (1,4) from directrix =25
The ratio of distance =520
From the above, we can conclude that the ratio is less than 1, so it is an ellipse.
Hence the required equation is (2(x+y))2(x−2)2+(y−1)2=(52)2
⇒2x2+y2+2xyx2+4−4x+y2+1−2y=54
⇒5x2+20−20x+5y2+5−10y=2x2+2y2+4xy
⇒3x2−4xy+3y2−20x−10y+25=0
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