Area of the greatest rectangle that can be inscribed in an ellipse \(\left(x^{2} / a^{2}\right)+\left(y^{2} / b^{2}\right)=1\) is (a) \(a b\) (b) \(2 \mathrm{ab}\) (c) \((\mathrm{a} / \mathrm{b})\) (d) \(\sqrt{(a b)}\)

Let's consider a rectangle of sides 2x and 2y inscribed inside the ellipse \(\left(x^{2} / a^{2}\right)+\left(y^{2} / b^{2}\right)=1\). We know that all the points on the ellipse satisfy the expression given. Therefore, since the corners of the rectangle lie on the ellipse, the expressions x and y must satisfy the ellipse equation. Now that we have defined that, let's express the relationship between x, y, a, and b: \[\left(\frac{x}{a}\right)^2+\left(\frac{y}{b}\right)^2=1\]

Since we need to find the area of the largest rectangle, we need to calculate the area of the given rectangle and maximize it. The area of the rectangle is given by: \[A = (2x)(2y) = 4xy\]

To find the maximum area, we can use calculus. To do this, first express y in terms of x using the ellipse equation: \[y = b\sqrt{1 - \frac{x^2}{a^2}}\] Now plug this expression for y into our area equation: \[A = 4x\left(b\sqrt{1 - \frac{x^2}{a^2}}\right)\] Now, we will find the critical points of A(x) by calculating the derivative with respect to x and setting it equal to zero: \[\frac{dA}{dx} = 4b\left(\sqrt{1 - \frac{x^2}{a^2}}\right) - \frac{8bx^3}{a^2\sqrt{1 - \frac{x^2}{a^2}}}\] Set this equal to zero and solve for x: \[4b\left(\sqrt{1 - \frac{x^2}{a^2}}\right) - \frac{8bx^3}{a^2\sqrt{1 - \frac{x^2}{a^2}}} = 0\] Simplifying this expression, we find that \(x = \frac{a}{\sqrt{2}}\). Now we can find the corresponding y value by substituting this x value into our ellipse equation: \[y = b\sqrt{1 - \frac{\left(\frac{a}{\sqrt{2}}\right)^2}{a^2}} = b\sqrt{1 - \frac{1}{2}} = \frac{b}{\sqrt{2}}\]

Finally, we can calculate the maximum area using the x and y values we found: \[A_{max} = 4\left(\frac{a}{\sqrt{2}}\right)\left(\frac{b}{\sqrt{2}}\right) = 2ab\] Therefore, the area of the greatest rectangle that can be inscribed in the given ellipse is \(2ab\), which corresponds to option (b).