The equation of the ellipse having foci (2, 0), ( -2, 0) and minor axis of length 8 units is:
1. \(\rm \frac{x^2}{16}+\frac{y^2}{20}=1\)
2. \(\rm \frac{x^2}{20}+\frac{y^2}{16}=1\)
3. \(\rm \frac{x^2}{2\sqrt5}+\frac{y^2}{4}=1\)
4. \(\rm \frac{x^2}{4}+\frac{y^2}{2\sqrt5}=1\)
Correct Answer – Option 2 : \(\rm \frac{x^2}{20}+\frac{y^2}{16}=1\)
Concept:
The distance between the centre and the focus of an ellipse is c = ae
The equation of an ellipse with the length of the major axis 2a and the minor axis 2b is given by: \(\rm \frac{x^2}{a^2}+\frac{y^2}{b^2}=1\).
Calculation:
Length of the minor axis = 2b = 8.
⇒ b = 4.
Also, c = distance between the centre and the focus = ae = 2.
c2 = a2e2 = a2 – b2
∴ 22 = a2 – 42
⇒ a2 = 20
Equation of the ellipse = \(\rm \frac{x^2}{a^2}+\frac{y^2}{b^2}=1\).
⇒ \(\rm \frac{x^2}{20}+\frac{y^2}{16}=1\).