If cosα = \textrm{ }\textrm{ } \frac{2}{3} , then the range of values of φ for which the point whose eccentric angle φ on the ellipse x2 + 4y2 = 4 falls inside the circle x2 + y2 + 4x + 3 = 0 is

Engineering

Mathematics

Standard Ellipse

Question

If $cosα = \frac{2}{3}$ , then the range of values of φ for which the point whose eccentric angle φ on the ellipse x² + 4y² = 4 falls inside the circle x² + y² + 4x + 3 = 0 is

(π – α, π)

(α, π)

(– α, α)

(0, α)

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Let point (2cos φ, sin φ)

4cos² φ + sin² φ + 8cos φ + 3 = 0

3cos² φ + 8cos φ + 4 < 0

(cos φ + 2) (3cos φ + 2) < 0

$cosϕ < \frac{- 2}{3}$

cos φ < cos (π – α)

φ > π – α.