Question

# The eccentricity of the conic $$4(2y-x-3)^{2}-9(2x+y-1)^{2}=80$$ is :

A
2
B
133
C
12
D
132

Solution

## The correct option is A $$\dfrac{\sqrt{13}}{3}$$The given equation can also be written as $$4\left(\dfrac{2y - x - 3}{\sqrt{5}}\right)^2 - 9\left(\dfrac{2x + y - 1}{\sqrt{5}}\right)^2 = 16$$Let $$X = \dfrac{2y - x - 3}{\sqrt{5}}$$ and $$Y = \dfrac{2x + y - 1}{\sqrt{5}}$$$$4X^2 - 9Y^2 = 16$$$$\Rightarrow \dfrac{X^2}{4} - \dfrac{Y^2}{\dfrac{16}{9}} = 1$$Now $$b^2 = a^2(e^2 - 1)$$So, $$\dfrac{16}{9} = 4(e^2 - 1)$$$$\Rightarrow e = \dfrac{\sqrt{13}}{3}$$Hence, option 'B' is correct.Maths

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