38. Which one of the points P = (3/2, 1/2), Q = (1/2, 3/2), R = (3/2, 11/2), and S = (11/2, 3/2) lies above the parabola y = 2x² and inside the circle x² + y² = 4?
(A) P
(B) Q
(C) R
(D) S
Find the Point That Lies Above the Parabola y = 2x² and Inside the Circle x² + y² = 4
Understanding the Conditions Given in the Question
This coordinate geometry problem requires us to identify a point that satisfies two conditions simultaneously. The required point must lie above the parabola y = 2x², and at the same time, it must lie inside the circle x² + y² = 4.
Therefore, checking only one of the two conditions is not sufficient. A point will be the correct answer only when it satisfies both inequalities associated with the given curves.
Condition for a Point to Lie Above the Parabola y = 2x²
The equation of the given parabola is:
y = 2x²
For a point (x, y) to lie exactly on this parabola, its coordinates must satisfy:
y = 2x²
However, the question asks for a point that lies above the parabola. Therefore, the y-coordinate of the point must be greater than the corresponding value of 2x².
Hence, the required condition is:
y > 2x²
We will substitute the coordinates of each point into this inequality to determine whether the point lies above the parabola.
Condition for a Point to Lie Inside the Circle x² + y² = 4
The equation of the given circle is:
x² + y² = 4
This circle has its centre at the origin (0, 0) and radius 2. A point lying exactly on the circle satisfies:
x² + y² = 4
A point lying inside the circle must satisfy:
x² + y² < 4
Therefore, for each given point, we need to calculate x² + y². If the result is less than 4, the point lies inside the circle.
Testing Point P = (3/2, 1/2)
Checking Whether P Lies Above the Parabola
For point P:
x = 3/2 and y = 1/2
Using the condition for the parabola:
2x² = 2(3/2)²
2x² = 2 × 9/4 = 9/2
The y-coordinate of point P is:
y = 1/2
Since:
1/2 < 9/2
the point P lies below the parabola, not above it. Therefore, point P does not satisfy the required condition.
Testing Point Q = (1/2, 3/2)
Checking Whether Q Lies Above the Parabola
For point Q:
x = 1/2 and y = 3/2
Substituting the x-coordinate into 2x²:
2x² = 2(1/2)²
2x² = 2 × 1/4 = 1/2
The y-coordinate of point Q is:
y = 3/2
Since:
3/2 > 1/2
the point Q lies above the parabola y = 2x². Thus, the first required condition is satisfied.
Checking Whether Q Lies Inside the Circle
Now, we check the second condition by calculating:
x² + y² = (1/2)² + (3/2)²
x² + y² = 1/4 + 9/4
x² + y² = 10/4 = 5/2
Since:
5/2 < 4
the point Q lies inside the circle x² + y² = 4. Therefore, point Q satisfies both required conditions.
Testing Point R = (3/2, 11/2)
Checking Whether R Lies Inside the Circle
For point R:
x = 3/2 and y = 11/2
Calculating x² + y²:
x² + y² = (3/2)² + (11/2)²
x² + y² = 9/4 + 121/4
x² + y² = 130/4 = 65/2
Since:
65/2 > 4
the point R lies outside the circle. Therefore, it cannot be the required point.
Testing Point S = (11/2, 3/2)
Checking Whether S Lies Inside the Circle
For point S:
x = 11/2 and y = 3/2
Calculating x² + y²:
x² + y² = (11/2)² + (3/2)²
x² + y² = 121/4 + 9/4
x² + y² = 130/4 = 65/2
Since:
65/2 > 4
the point S lies outside the circle. Therefore, point S also fails to satisfy the required conditions.
Comparing All the Given Points
Point P lies below the parabola, so it is not the correct answer. Points R and S lie outside the circle, so they are also eliminated. Point Q satisfies both conditions because it lies above the parabola and inside the circle.
For point Q = (1/2, 3/2):
3/2 > 2(1/2)² = 1/2
Therefore, Q lies above the parabola.
Also:
(1/2)² + (3/2)² = 5/2 < 4
Therefore, Q lies inside the circle.
Final Answer
The point that lies above the parabola y = 2x² and inside the circle x² + y² = 4 is:
Q = (1/2, 3/2)
Correct Option: (B) Q
