Ellipse given the foci and a point

Problem

Write the equation of the ellipse with foci $F_{1,2}(\pm 3;0)$ and passing through $P(4;0)$.

Solution

The foci are on the $x$-axis and symmetric with respect to the origin, hence $c = 3$ and the ellipse is in canonical form $\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1$.

The point $P(4;0)$ has zero ordinate: it is therefore a vertex on the $x$-axis, which means $a = 4$.

From the relation $b^2 = a^2 - c^2$: $b^2 = 16 - 9 = 7.$

The required equation is $\boxed{\dfrac{x^2}{16} + \dfrac{y^2}{7} = 1}.$

Links

Topics: Ellipse
Concepts: Ellipse · Canonical equation · Foci · Semi-axes
Skills: Analytic geometry · Using formulae
Exercise type: Geometric problem