Find the equation of the ellipse which passes through the point, [4,1] and having its foci at [+-3,0] - Maths - Introduction to Three Dimensional Geometry

Question

Find the equation of the ellipse which passes through the point,
[4,1] and having its foci at [+-3,0]

ramandeep.singh · Accepted Answer

Let the equation of the ellipse be

$\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$

Let e be the eccentricity.

As the ellipse passes through the point (4, 1), therefore, (4, 1) must satisfy the equation of the ellipse.

$∴ \frac{4^{2}}{a^{2}} + \frac{1^{2}}{b^{2}} = 1 \Rightarrow \frac{16}{a^{2}} + \frac{1}{b^{2}} = 1....... (1)$

The foci are given as (±3, 0).

∴ae = ±3

We know that

$b^{2} = a^{2} (1 - e^{2}) = a^{2} - (a e)^{2} \Rightarrow b^{2} = a^{2} - 9...... (2)$

Substituting the value of $b^{2}$ from equation (2) in equation (1), we get

$a^{2} = 8 o r a^{2} = 18$

But $a^{2}$ can't have the value 8, as then $b^{2}$ will be negative, therefore, we have

$a^{2} = 18 ∴ b^{2} = 18 - 9 = 9$

Thus, the equation of the ellipse is given by

$\frac{x^{2}}{18} + \frac{y^{2}}{9} = 1$

Find the equation of the ellipse which passes through the point, [4,1] and having its foci at [+-3,0].