Question

The major and minor axis of the ellipse $400x^{2}+100y^{2}=40000$ respectively are

• A. 100 and 20
• B. 20 and 10
• C. 40 and 20
• D. 400 and 100
• E. 16 and 8

Hint:

Geometry Tip: Always remember that $a$ and $b$ represent the distance from the centre to the edge (the "radius" of the ellipse). The question asks for the full axis lengths, so you must always double them to get $2a$ and $2b$.

Answer 1

The Correct Option is C

Concept:
The standard form of an ellipse equation is $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$. The length of the axes depends on the denominators $a^2$ and $b^2$. The longer axis is the "major axis" with length $2 \times \max(a, b)$, and the shorter axis is the "minor axis" with length $2 \times \min(a, b)$.

Step 1: Convert the equation to standard form.
The given equation is $400x^2 + 100y^2 = 40000$. To make the right side equal to 1, divide the entire equation by 40000: $$\frac{400x^2}{40000} + \frac{100y^2}{40000} = \frac{40000}{40000}$$

Step 2: Simplify the fractions.
Reduce the fractions to identify the denominators: $$\frac{x^2}{100} + \frac{y^2}{400} = 1$$ This matches the standard form $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$.

Step 3: Identify the semi-axes parameters a and b.
From the simplified equation: $$a^2 = 100 \implies a = 10$$ $$b^2 = 400 \implies b = 20$$

Step 4: Determine the major axis length.
Because $b > a$, the ellipse is oriented vertically. The semi-major axis is $b = 20$. The full length of the major axis is $2b$: $$\text{Major Axis} = 2(20) = 40$$

Step 5: Determine the minor axis length.
The semi-minor axis is $a = 10$. The full length of the minor axis is $2a$: $$\text{Minor Axis} = 2(10) = 20$$ The lengths are 40 and 20 respectively. Hence the correct answer is (C) 40 and 20.