Question

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