Conic Sections: Conceptual Doubts Cleared (12)

Question

Find the equation of the ellipse whose foci are $(\pm 4, 0)$ and the length of the major axis is $10$. Also find its eccentricity and the equations of its directrices.

Solution — Step by Step

Final Answer: Ellipse: $\dfrac{x^2}{25} + \dfrac{y^2}{9} = 1$, $e = 4/5$, directrices $x = \pm 25/4$.

Why This Works

For an ellipse, $a$ is the semi-major axis, $c$ is the focal distance from centre, and $b^2 = a^2 - c^2$ comes from the defining property “sum of distances to the two foci is constant ($= 2a$)”. These three numbers fully determine the ellipse if the centre is at the origin.

The eccentricity $e = c/a$ measures how “stretched” the ellipse is: $e = 0$ is a circle, $e \to 1$ is a very elongated ellipse, and $e = 1$ becomes a parabola.

Alternative Method

Use the focus-directrix definition: for any point $P$ on the ellipse, distance to focus $/$ distance to directrix $= e$. Plug in a known vertex to verify: at $(5, 0)$, distance to focus $(4, 0)$ is $1$, distance to directrix $x = 25/4$ is $25/4 - 5 = 5/4$. Ratio $= 1 / (5/4) = 4/5 = e$. ✓