(A) P
(B) Q
(C) R
(D) S
Find the Point Above the Parabola y = 2x² and Inside the Circle x² + y² = 4
Understanding the Coordinate Geometry Problem
This question asks us to identify one point that satisfies two geometrical conditions simultaneously. The point must lie above the parabola y = 2x2, and at the same time, it must lie inside the circle x2 + y2 = 4.
The important word is and. A point is correct only when both conditions are satisfied. A point that lies above the parabola but outside the circle is not acceptable. Similarly, a point inside the circle but below the parabola is also not acceptable.
Therefore, for every given point (x, y), we need to check the following two inequalities:
Condition 1: y > 2x2
and:
Condition 2: x2 + y2 < 4
The point that satisfies both inequalities is the required answer.
Condition for a Point to Lie Above the Parabola y = 2x²
The equation of the given parabola is:
y = 2x2
For any fixed value of x, the corresponding point on the parabola has y-coordinate 2x2. A point lies above the parabola when its actual y-coordinate is greater than this value.
Therefore, the condition for a point (x, y) to lie above the parabola is:
y > 2x2
If y < 2x2, the point lies below the parabola. Thus, each option can be tested by comparing its y-coordinate with the value of 2x2.
Condition for a Point to Lie Inside the Circle x² + y² = 4
The given circle is:
x2 + y2 = 4
This is a circle centered at the origin (0, 0) with radius 2 because the standard equation of a circle centered at the origin is:
x2 + y2 = r2
Here:
r2 = 4
Therefore:
r = 2
A point lies inside this circle when its squared distance from the origin is less than 4. Hence, the required condition is:
x2 + y2 < 4
If x2 + y2 is greater than 4, the point lies outside the circle.
Step-by-Step Analysis of All Options
Option (A): P = (3/2, 1/2)
For point P, the coordinates are:
x = 3/2, y = 1/2
First, we test whether P lies above the parabola. The value of 2x2 is:
2x2 = 2(3/2)2
= 2 × 9/4
= 9/2
The y-coordinate of P is only 1/2. Therefore:
1/2 < 9/2
Thus, point P lies below the parabola, not above it.
For completeness, we can also test the circle condition:
x2 + y2 = (3/2)2 + (1/2)2
= 9/4 + 1/4
= 10/4 = 5/2
Since 5/2 < 4, point P lies inside the circle. However, it fails the parabola condition. Therefore, Option (A) is incorrect.
Option (B): Q = (1/2, 3/2)
For point Q, the coordinates are:
x = 1/2, y = 3/2
First, calculate the value of the parabola at x = 1/2:
2x2 = 2(1/2)2
= 2 × 1/4
= 1/2
The actual y-coordinate of Q is 3/2. Therefore:
3/2 > 1/2
Hence, point Q lies above the parabola.
Now test whether Q lies inside the circle:
x2 + y2 = (1/2)2 + (3/2)2
= 1/4 + 9/4
= 10/4
= 5/2
Since:
5/2 < 4
point Q also lies inside the circle.
Therefore, Q satisfies both required conditions:
Q lies above the parabola and inside the circle.
Hence, Option (B) is correct.
Option (C): R = (3/2, 11/2)
For point R:
x = 3/2, y = 11/2
For the parabola:
2x2 = 2(3/2)2 = 9/2
Since:
11/2 > 9/2
point R lies above the parabola.
However, for the circle:
x2 + y2 = (3/2)2 + (11/2)2
= 9/4 + 121/4
= 130/4
= 65/2
Since 65/2 is much greater than 4, point R lies outside the circle.
Therefore, R satisfies the parabola condition but fails the circle condition. Hence, Option (C) is incorrect.
Option (D): S = (11/2, 3/2)
For point S:
x = 11/2, y = 3/2
Testing the parabola condition:
2x2 = 2(11/2)2
= 2 × 121/4
= 121/2
The y-coordinate is only 3/2. Therefore:
3/2 < 121/2
Thus, point S lies below the parabola.
It also lies outside the circle because:
x2 + y2 = (11/2)2 + (3/2)2
= 121/4 + 9/4
= 130/4
= 65/2 > 4
Therefore, point S satisfies neither of the required conditions. Hence, Option (D) is incorrect.
Comparison of All Four Points
| Point | Above y = 2x2? | Inside x2 + y2 = 4? | Result |
|---|---|---|---|
| P = (3/2, 1/2) | No | Yes | Incorrect |
| Q = (1/2, 3/2) | Yes | Yes | Correct |
| R = (3/2, 11/2) | Yes | No | Incorrect |
| S = (11/2, 3/2) | No | No | Incorrect |
Why Point Q Is the Only Correct Point
Point Q = (1/2, 3/2) is the only point that satisfies both inequalities simultaneously.
For the parabola:
y = 3/2 > 2(1/2)2 = 1/2
Therefore, Q lies above the parabola.
For the circle:
(1/2)2 + (3/2)2 = 5/2 < 4
Therefore, Q lies inside the circle.
Since both conditions are true, Q is the required point.
Geometrical Interpretation of the Answer
The parabola y = 2x2 opens upward and has its vertex at the origin. The region above the parabola consists of points whose y-coordinate is greater than 2x2.
The circle x2 + y2 = 4 is centered at the origin and has radius 2. The required point must therefore lie within the circular region while also remaining above the upward-opening parabola.
Point Q = (1/2, 3/2) lies close enough to the origin to remain inside the circle, and its y-coordinate is sufficiently large compared with its x-coordinate to place it above the parabola.
The other points fail at least one of these two geometrical conditions.
Final Answer
The point Q = (1/2, 3/2) lies above the parabola y = 2x2 and inside the circle x2 + y2 = 4.
Correct Option: (B) Q
