Mathematics - Area Under Curves Question with Solution | TestHub

The given equation of the ellipse is $\frac{x^2}{25}+\frac{y^2}{9}=1$.

Comparing with the standard form $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$, we get $a=5$ and $b=3$.

The area of the ellipse in the first quadrant (bounded by the arc AB and the coordinate axes) is given by $\frac{1}{4}\pi ab$.

Area of the quarter ellipse $=\frac{1}{4}\pi (5)(3)=\frac{15\pi }{4}$ sq. units.

The points A and B are the intercepts on the positive x and y axes, respectively, so $A=(5,0)$ and $B=(0,3)$.

The area of the right-angled triangle OAB formed by the chord AB and the coordinate axes is $\frac{1}{2}\times OA\times OB$.

Area of $\Delta OAB=\frac{1}{2}\times 5\times 3=\frac{15}{2}$ sq. units.

The required area between the arc AB and the chord AB is the difference between the area of the quarter ellipse and the area of the triangle OAB.

Required area $=\frac{15\pi }{4}-\frac{15}{2}=\frac{15}{4}(\pi -2)$ sq. units.

Answer: $\frac{15}{4}(\pi -2)$