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Mathematics · 2026 · 4 marks
CBSE 2026 · Region 2 · Set 1 · Q37
A racing track is build around an elliptical ground whose equation is given by 9 x^2+16 y^2=144. The width of the track is 3 m as shown below: Based…
A racing track is build around an elliptical ground whose equation is given by $\displaystyle 9 x^{2}+16 \mathrm{y}^{2}=144$. The width of the track is $\displaystyle 3$ m as shown below :
Based on given information, answer the following questions :
(i)
Express y as a function of $\displaystyle x$ from the given equation of ellipse.
(ii)
Integrate the function obtained in (i) with respect to $\displaystyle x$.
(iii)
(a)
Find the area of the region enclosed within the elliptical ground excluding the track using integration.
OR
(b)
Write the co-ordinates of the points P and Q where the outer edge of the track cuts $\displaystyle x$ axis and y axis in first quadrant and find the area of the triangle formed by points $\displaystyle \mathrm{P}, \mathrm{O}, \mathrm{Q}$ using integration.
Marking-scheme solution
(i)
$\displaystyle 9 x^{2}+16 y^{2}=144 \Rightarrow y=\dfrac{3}{4} \sqrt{16-x^{2}}$
(ii)
$\displaystyle \int y\, d x=\int \dfrac{3}{4} \sqrt{16-x^{2}}\, d x$
$\displaystyle =\dfrac{3}{4}\left[\dfrac{x}{2} \sqrt{16-x^{2}}+8 \sin^{-1}\left(\dfrac{x}{4}\right)\right]+C$
(iii)
Required Area $\displaystyle =4 \times \int_{0}^{4} \dfrac{3}{4} \sqrt{16-x^{2}}\, d x$
$\displaystyle =3\left[\dfrac{x}{2} \sqrt{16-x^{2}}+8 \sin^{-1}\left(\dfrac{x}{4}\right)\right]_{0}^{4}$
$\displaystyle =3\left(8 \sin^{-1} 1-0\right)=24 \times \dfrac{\pi}{2}=12 \pi$
The track has a width of $\displaystyle 3$ m. Clearly $\displaystyle P(7,0)$ and $\displaystyle Q(0,6)$
The equation of line passing through $\displaystyle P(7,0)$ and $\displaystyle Q(0,6)$ is $\displaystyle \dfrac{x}{7}+\dfrac{y}{6}=1$
Here, $\displaystyle y=\dfrac{6}{7}(7-x)$
Required area $\displaystyle =\int_{0}^{7} y\, d x=\int_{0}^{7} \dfrac{6}{7}(7-x)\, d x$
$\displaystyle =\dfrac{6}{7}\left[\dfrac{(7-x)^{2}}{-2}\right]_{0}^{7}=21$
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CBSE Class 12 Mathematics past-paper question from the 2026 board exam, with the answer as CBSE’s own marking scheme gives it. Where our answers come from.