QJEEM17Ma5187: JEE Main 2017 Mathematics - Class 11 JEEM17Ma5187 Solution: The eccentricity of an ellipse having centre at the origin,…

Centre at (0, 0)

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

at point (4, $$-$$ 1)

$$\frac{16}{a^2}+\frac{1}{b^2}$$ = 1

$$\Rightarrow$$   16b² + a² = a²b²         . . . .(i)

at point ($$-$$ 2, 2)

$$\frac{4}{a^2}+\frac{4}{b^2}=1$$

$$\Rightarrow$$   4b² + 4a² = a²b²          . . . .(ii)

$$\Rightarrow$$   16b² + a² = 4a² + 4b²

From equations (i) and (ii)

$$\Rightarrow$$   3a² = 12b²

$$\Rightarrow$$   a² = 4b²

b² = a²(1 $$-$$ e²)

$$\Rightarrow$$   e² = $$\frac{3}{4}$$

$$\Rightarrow$$   e = $$\frac{\sqrt{3}}{2}$$