Question

The curve represented by the equation $ 4{{x}^{2}}+16{{y}^{2}}-24x-32y-12=0 $ is:

• A. a parabola
• B. a pair of straight lines
• C. an ellipse with eccentricity 1/2
• D. an ellipse with eccentricity $ \sqrt{3}/2 $
• E. a hyperbola with eccentricity 3/2

Answer 1

Answer: (D)

The given equation is $ 4{{x}^{2}}-24x+36+16{{y}^{2}}-32y-12-36 $ $ +16-16=0 $ $ \Rightarrow $ $ {{(2x-6)}^{2}}+{{(4y-4)}^{2}}=64 $ $ \Rightarrow $ $ \frac{{{(x-3)}^{2}}}{16}+\frac{{{(y-1)}^{2}}}{4}=1 $ This represents an ellipse and $ {{a}^{2}}=16,\text{ }{{b}^{2}}=4 $ $ \therefore $ $ e=\sqrt{1-\frac{4}{16}}=\frac{\sqrt{3}}{2} $