Q.12. A family of quadratic curves is given by

$$y_k = 2 \left( x - \frac{k}{2} \right)^2 + \frac{k^2}{2} + 4k + 3$$

where $k$ is any real number and $y_k$ is a function of $x$.

All these curves are sketched, and the point with the lowest $y$-coordinate among all the curves $y_k$ is $(a, b)$.

Find the value of $a + b$.

Answer Choices

• $-1$
• $-3$
• $-5$
• $-7$
• $-9$

Solution

We are given the quadratic function in completed square form. The minimum $y$-coordinate for each curve occurs when $x = \frac{k}{2}$, and the value of $y_k$ at this point is $\frac{k^2}{2} + 4k + 3$.

Minimizing this quadratic expression for $k$, we find that the minimum occurs at $k = -4$, and the corresponding minimum point is $(-2, -5)$.

Thus, the value of $a + b = -2 + (-5) = -7$.

The correct answer is: $-7$ .