90. A circle is given by the equation
2x² + 2y² + 8x − 20y + 10 = 0.
The area of a square whose side equals the radius of the circle is ______.
Finding the Radius of a Circle from Its General Equation
The present question asks for the radius of a circle given in its general form. The standard approach involves converting the equation into the standard form of a circle by using the method of completing the square. Once the equation is written in standard form, the centre and radius can be identified immediately. This technique is one of the most important algebraic tools in coordinate geometry and is frequently tested in competitive examinations.
Correct Answer
36
Understanding the Concept
The general equation of a circle is usually written as
x² + y² + 2gx + 2fy + c = 0.
To identify the centre and radius, the equation must first be transformed into the standard form
(x − h)² + (y − k)² = r².
Here,
- (h, k) is the centre of the circle.
- r is the radius.
Since the coefficients of x² and y² in the given equation are both equal to 2, we first divide the entire equation by 2.
Step 1: Simplify the Equation
Given equation:
2x² + 2y² + 8x − 20y + 10 = 0
Divide every term by 2.
x² + y² + 4x − 10y + 5 = 0
Step 2: Group the x and y Terms
Rearrange the equation as
(x² + 4x) + (y² − 10y) = −5
Now complete the square for both groups.
Step 3: Complete the Square
For the x terms:
x² + 4x = (x + 2)² − 4
For the y terms:
y² − 10y = (y − 5)² − 25
Substituting these expressions into the equation gives
(x + 2)² − 4 + (y − 5)² − 25 = −5
Simplifying,
(x + 2)² + (y − 5)² = 24
Step 4: Compare with the Standard Equation
The standard equation of a circle is
(x − h)² + (y − k)² = r²
Comparing with
(x + 2)² + (y − 5)² = 24
we obtain
- Centre = (−2, 5)
- Radius² = 24
Therefore,
Radius = √24 = 2√6.
Step 5: Find the Area of the Square
The question states that the side of the square is equal to the radius of the circle.
Hence,
Side = 2√6.
The area of a square is
Side².
Therefore,
Area = (2√6)²
= 4 × 6
= 24.
Mathematical Verification
Radius² = 24 ✔
Radius = 2√6 ✔
Area of square = (2√6)² ✔
= 24 ✔
The computation is completely verified.
Important Observation
Using the given equation exactly as printed, the radius of the circle is 2√6, and therefore the area of the square is 24.
If an official answer key gives 36, then the equation in the question is likely to contain a typographical error. For example, if the constant term were different, the radius would change accordingly.
With the equation shown in the image, the mathematically correct value is 24.
Explanation of the Method
Completing the square is the standard technique used whenever a circle is given in its general equation. By converting the quadratic expression into perfect squares, the equation immediately reveals the centre and the radius. This method is not only useful for circles but also forms the basis for studying ellipses, parabolas, and hyperbolas in coordinate geometry.
Alternative Formula Method
For the equation
x² + y² + 2gx + 2fy + c = 0,
the radius is given by
r = √(g² + f² − c).
Here,
g = 2
f = −5
c = 5
Therefore,
r² = 2² + (−5)² − 5
= 4 + 25 − 5
= 24.
This confirms the previous result without completing the square.
Related Practice Example
Find the radius of the circle
x² + y² − 6x + 8y + 9 = 0.
Using the radius formula,
g = −3,
f = 4,
c = 9.
Thus,
r² = (−3)² + 4² − 9
= 9 + 16 − 9
= 16.
Hence,
r = 4.
This example reinforces the shortcut formula used in the present problem.
Key Takeaways
Whenever the equation of a circle is given in its general form, first divide by the common coefficient of x² and y² if necessary. Then either complete the square or apply the direct radius formula. Finally, use the radius to answer the quantity asked in the problem, such as area, circumference, or diameter.
Final Answer
The given equation simplifies to
(x + 2)² + (y − 5)² = 24.
Therefore,
Radius = 2√6.
The side of the square equals the radius, so
Area = (2√6)² = 24.
Final Answer: 24
