If the tangents drawn from a point P to the ellipse 4 x²+9 y²-16 x+54 y+61=0 are perpendicular, then the locus of P is [AP EAPCET 23 May 2025 S1]

Correct Answer is : x2+y2−4x+6y=0x²+y²-4 x+6 y=0x2+y2−4x+6y=0

Correct Answer is : x2+y2−4x+6y=0x²+y²-4 x+6 y=0x2+y2−4x+6y=0

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Section: Mathematics

Question : 3 of 16

Marks: +1, -0

P

4 x^2+9 y^2-16 x+54 y+61=0

Solution:

\;4 x^2+9 y^2-16 x+54 y+61=0

\; \Rightarrow \;\; (2 x-4)^2+(3 y+9)^2=36

\; \Rightarrow \;\; \frac{(x-2)^2}{9}+\frac{(y+3)^2}{4}=1

Which is of the form

\;\frac{x^2}{a^2}+\frac{y^2}{b^2}=1

where

X=x-2 Y=y+3, a=3

and

b=2

As we know equation of director circle (because locus of two perpendicular pair of tangents shows its director circle) is

x^2+y^2=a^2+b^2

\; \Rightarrow \;\; (x-2)^2+(y+3)^2=(3)^2+(2)^2

\; \Rightarrow \;\; x^2+4-4 x+y^2+9+6 y=13

\; \Rightarrow \;\; x^2+y^2-4 x+6 y=0

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