Question

Which one of the points P = \(\left(\frac{3}{2}, \frac{1}{2}\right)\), Q = \(\left(\frac{1}{2}, \frac{3}{2}\right)\), R = \(\left(\frac{3}{2}, \frac{11}{2}\right)\) and S = \(\left(\frac{11}{2}, \frac{3}{2}\right)\) lies ABOVE the parabola \(y = 2x^2\) and INSIDE the circle \(x^2 + y^2 = 4\)?

• A. P
• B. Q
• C. R
• D. S

Hint:

To check if a point lies above a parabola and inside a circle, evaluate its coordinates in the given equations and verify both conditions.

Answer 1

The Correct Option is B

Step 1: Analyzing the parabola and circle equations.
We are given a parabola \(y = 2x^2\) and a circle \(x^2 + y^2 = 4\). To find the points that lie above the parabola and inside the circle, we need to check each point’s coordinates against these two conditions.
Step 2: Check the points.
- For \(P = \left(\frac{3}{2}, \frac{1}{2}\right)\), check if it satisfies both conditions:
- Parabola: \(y = 2x^2 \Rightarrow \frac{1}{2} \neq 2\left(\frac{3}{2}\right)^2 = \frac{9}{2}\), so \(P\) does not satisfy the parabola condition.
- Circle: \(x^2 + y^2 = \left(\frac{3}{2}\right)^2 + \left(\frac{1}{2}\right)^2 = \frac{9}{4} + \frac{1}{4} = 2.5\), which is inside the circle.
- For \(Q = \left(\frac{1}{2}, \frac{3}{2}\right)\), check both conditions:
- Parabola: \(y = 2x^2 \Rightarrow \frac{3}{2} = 2\left(\frac{1}{2}\right)^2 = \frac{1}{2}\), so it satisfies the parabola condition.
- Circle: \(x^2 + y^2 = \left(\frac{1}{2}\right)^2 + \left(\frac{3}{2}\right)^2 = \frac{1}{4} + \frac{9}{4} = 2.5\), which is inside the circle.
Step 3: Conclusion.
The correct answer is (B) Q because \(Q\) lies both above the parabola and inside the circle.