Find e = √(1 - b²/a²) and identify major/minor axes.

Eccentricity of Ellipse x²/25 + y²/16 = 1

Question

Find the eccentricity of the ellipse $\dfrac{x^2}{25} + \dfrac{y^2}{16} = 1$ . Also identify the major and minor axes.

Solution — Step by Step

Read directly from the standard form $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$ . Here $a^2 = 25$ and $b^2 = 16$ , so $a = 5$ and $b = 4$ .

Since $a^2 > b^2$ (25 > 16), the larger denominator is under $x^2$ . This means the major axis lies along the x-axis, with length $2a = 10$ . The minor axis is along the y-axis, with length $2b = 8$ .

We need $c$ first because eccentricity $e = c/a$ . The relation $c^2 = a^2 - b^2$ comes from the geometric definition of an ellipse — $c$ is the distance from the centre to each focus.

c^2 = 25 - 16 = 9 \implies c = 3

e = \frac{c}{a} = \frac{3}{5} = \mathbf{0.6}

For any ellipse, $0 < e < 1$ . We got $e = 0.6$ — sits comfortably in that range. ✓

Why This Works

The eccentricity formula $e = \sqrt{1 - \dfrac{b^2}{a^2}}$ measures how “stretched” the ellipse is. When $e \to 0$ , the ellipse becomes a perfect circle. When $e \to 1$ , it flattens into a line segment.

Here $e = 0.6$ tells us the ellipse is moderately elongated — not too circular, not too flat. The foci are at $(\pm 3, 0)$ , sitting inside the ellipse between the centre and the vertices at $(\pm 5, 0)$ .

The key insight is that $a$ is always the larger of the two values. Many students blindly write $a^2 = 25$ without checking which axis is major — this matters when writing focus coordinates and directrix equations.

Alternative Method

We can use the eccentricity formula directly without finding $c$ separately:

e = \sqrt{1 - \frac{b^2}{a^2}} = \sqrt{1 - \frac{16}{25}} = \sqrt{\frac{9}{25}} = \frac{3}{5}

This single-step formula is faster in JEE Main where time is tight. Memorise it as: subtract the fraction of denominators, take square root. Two lines, done.

Common Mistake

Swapping a² and b² — Students write $a^2 = 16$ because 16 comes first alphabetically or they confuse rows. Always check: $a^2$ is the larger denominator for a standard ellipse. If you had written $a^2 = 16, b^2 = 25$ , you’d get $c^2 = 16 - 25 < 0$ , which is impossible. That negative under the square root is your signal something went wrong.

This question appeared in JEE Main 2024 and is a guaranteed 4-marker in CBSE Class 11 boards. The calculation is short, but the marks go to students who correctly state the major axis direction — examiners specifically check this.

Question

Solution — Step by Step

Why This Works

Alternative Method

Common Mistake

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