Question

1. write an equation of the ellipse with foci (±4, 0) and covertices at (0, ±2)

options:

1. $\frac{x^2}{2} + \frac{y^2}{20} = 1$
2. $\frac{x^2}{20} + \frac{y^2}{4} = 1$
3. $\frac{x^2}{20} + \frac{y^2}{2} = 1$
4. $\frac{x^2}{4} + \frac{y^2}{20} = 1$

Explanation:

Step1: Identify given values

Foci: $(\pm 4, 0)$, so $c=4$; co-vertices: $(0, \pm 2)$, so $b=2$

Step2: Calculate $a^2$ using $c^2=a^2-b^2$

$a^2 = c^2 + b^2 = 4^2 + 2^2 = 16 + 4 = 20$

Step3: Write ellipse equation (horizontal major axis)

Standard form: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$, substitute values: $\frac{x^2}{20} + \frac{y^2}{4} = 1$

Answer:

$\boldsymbol{\frac{x^2}{20} + \frac{y^2}{4} = 1}$ (corresponding to the third option)