Sketch the curve \(x, y) : 100x² + 25y² = 2500\ and find the area of the region enclosed by it, using integration.

MathsCBSE Class 12 · Board exam 2026, set 65/5/1 · 5 marks · Long answer

CBSE Class 12 Mathematics 2026: question 34

Set in the CBSE Class 12 board exam of 2026, paper 65/5/1, section D, question 34, for 5 marks. Try it first; the hints and our step-by-step solution are below.

Q34 · 5 marks · Long answerApplication of Integrals

Sketch the curve ${(x, y) : 100 x^{2} + 25 y^{2} = 2500}$ and find the area of the region enclosed by it, using integration.

Source: official question paper from cbse.gov.in (PDF). See it in the full 2026 board paper, set 65/5/1.

Hints

Open one hint at a time and try again before the next.

Hint 1

Divide the equation by 2500 to get the standard form of an ellipse and read off its intercepts.

Hint 2

Use symmetry: find the area in the first quadrant and multiply by 4.

Hint 3

In the first quadrant $y = 2 25 - x^{2}$ ; use the formula for $\int a^{2} - x^{2} d x$ .

Step-by-step solution

Each step shows the marks it earns, as an examiner would award them. Checked by an independent re-solve from the question paper.

Divide by 2500: $\frac{x ^{2}}{25} + \frac{y ^{2}}{100} = 1$ . This is an ellipse centred at the origin with major axis along the y-axis. It meets the x-axis at $(\pm 5, 0)$ and the y-axis at $(0, \pm 10)$ .
Sketch: draw a vertical ellipse through $(5, 0), (- 5, 0), (0, 10), (0, - 10)$ and shade the region inside it.
1½ marks
In the first quadrant $y = \frac{10}{5} 25 - x^{2} = 2 25 - x^{2}$ . The ellipse is symmetric about both axes, so
Required area $= 4 \int_{0}^{5} 2 25 - x^{2} d x = 8 \int_{0}^{5} 25 - x^{2} d x$ .
1½ marks
Using $\int a^{2} - x^{2} d x = \frac{x}{2} a^{2} - x^{2} + \frac{a ^{2}}{2} sin^{- 1} \frac{x}{a}$ :
$= 8 [\frac{x}{2} 25 - x^{2} + \frac{25}{2} sin^{- 1} \frac{x}{5}]_{0}^{5}$
1 mark
$= 8 [(0 + \frac{25}{2} \cdot \frac{π}{2}) - (0 + 0)] = 8 \cdot \frac{25 π}{4} = 50 π$ sq. units.
1 mark

Answer: Area $= 50 π$ sq. units

Where the marks are

Writes the ellipse in standard form $\frac{x ^{2}}{25} + \frac{y ^{2}}{100} = 1$ and draws a correct sketch (1½ marks)
Sets up the area as $4 \int_{0}^{5} 2 25 - x^{2} d x$ (1½ marks)
Integrates correctly using the standard formula (1 mark)
Obtains the area $50 π$ sq. units (1 mark)

Revise the chapter

Chapter: Application of Integrals (notes, formulas, where marks are lost and every past question)
Topic: Area of Circle and Ellipse by Integration

Practise this chapterBoard-style questions from Application of Integrals, one at a time, marked against the answer key and marking points.Back to the full paper2026 board paper, set 65/5/1: every question, by section.

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CBSE Class 12 Mathematics 2026: question 34

Hints

Step-by-step solution

Revise the chapter

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