Question

Let the parabola \( y = x^2 + px + q \) passing through the point \( (1, -1) \) be such that the distance between its vertex and the x-axis is minimum. Then the value of \( p^2 + q^2 \) is:

• A. 2
• B. 4
• C. 5
• D. 8

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
The vertex of a parabola given by \(y = ax^2 + bx + c\) is located at \(x = \frac{-b}{2a}\).
The distance from the vertex to the \(x\)-axis is simply the absolute value of the \(y\)-coordinate of the vertex.
We are given that the parabola passes through a specific point, which gives us a relation between \(p\) and \(q\).
Step 2: Key Formula or Approach:
For the parabola \(y = x^2 + px + q\), the \(x\)-coordinate of the vertex is \(x_v = \frac{-p}{2}\).
The \(y\)-coordinate of the vertex is \(y_v = \left(\frac{-p}{2}\right)^2 + p\left(\frac{-p}{2}\right) + q = \frac{-p^2}{4} + q\).
The condition that the parabola passes through \((1, -1)\) means \(-1 = 1^2 + p(1) + q\).
Step 3: Detailed Explanation:
From the passing point condition, we can express \(q\) in terms of \(p\).
\[ -1 = 1 + p + q \implies p + q = -2 \implies q = -2 - p \] Substitute this expression for \(q\) into the \(y\)-coordinate of the vertex.
\[ y_v = \frac{-p^2}{4} + (-2 - p) = -\left(\frac{p^2}{4} + p + 2\right) \] The distance from the vertex to the \(x\)-axis is \(D = |y_v|\).
\[ D = \left| \frac{p^2}{4} + p + 2 \right| \] To minimize this distance, we complete the square for the quadratic expression inside the absolute value.
\[ D = \frac{1}{4} \left| p^2 + 4p + 8 \right| \] \[ D = \frac{1}{4} \left| (p^2 + 4p + 4) + 4 \right| \] \[ D = \frac{1}{4} \left| (p + 2)^2 + 4 \right| \] Since \((p + 2)^2 \ge 0\), the minimum value of the expression inside the absolute value occurs when \((p + 2)^2 = 0\).
This happens when \(p = -2\).
With \(p = -2\), we find the corresponding value of \(q\).
\[ q = -2 - (-2) = 0 \] We need to find the value of \(p^2 + q^2\).
\[ p^2 + q^2 = (-2)^2 + 0^2 = 4 + 0 = 4 \] Step 4: Final Answer:
The value of \(p^2 + q^2\) is \(4\).