Answer
$$\langle u, v \rangle= \frac{1}{25}u_1v_1+\frac{1}{4}u_2v_2.$$
Work Step by Step
The figure shows an ellipse centered at the origin and its equation is given by
$$\frac{x^2}{5^2}+\frac{y^2}{2^2}=1 \quad \quad(1)$$
Now, since $\|u\|=1$, then
$$\|u\|^2 = \langle u, u \rangle= c_1u_1^2+c_2u_2^2, \quad \quad (2).$$
Comparing (1) and (2), we get
$$c_1=\frac{1}{25}, \quad c_2=\frac{1}{4}.$$
Consequently, the inner product is given by
$$\langle u, v \rangle= \frac{1}{25}u_1v_1+\frac{1}{4}u_2v_2.$$
