Question #195309

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
2021-05-21T07:38:15-0400

Given,

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

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

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

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

The distance from the point (1,4)(1,4) from directrix =52=\frac{5}{\sqrt{2}}

The ratio of distance =205=\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 (x2)2+(y1)2((x+y)2)2=(25)2\frac{(x-2)^2+(y-1)^2}{(\frac{(x+y)}{\sqrt{2}})^2}=(\frac{2}{\sqrt{5}})^2

x2+44x+y2+12yx2+y2+2xy2=45\Rightarrow \frac{x^2+4-4x+y^2+1-2y}{\frac{x^2+y^2+2xy}{2}}=\frac{4}{5}

5x2+2020x+5y2+510y=2x2+2y2+4xy\Rightarrow 5x^2+20-20x+5y^2+5-10y=2x^2+2y^2+4xy

3x24xy+3y220x10y+25=0\Rightarrow 3x^2-4xy+3y^2-20x-10y+25=0


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