The equation of a circle is 4x2 + 4y 2− 24x − 32y - 96 = 0. Write the equation in standard form.
Question
The equation of a circle is
Write the equation in standard form.
Solution
The standard form of a circle's equation is (x-h)² + (y-k)² = r², where (h,k) is the center of the circle and r is the radius.
Given the equation 4x² + 4y² - 24x - 32y - 96 = 0, we can rewrite it as:
4(x² - 6x) + 4(y² - 8y) = 96
Now, to complete the square, we take half of the coefficients of x and y, square them and add to both sides. Half of -6 is -3, and (-3)² = 9. Half of -8 is -4, and (-4)² = 16.
So, we rewrite the equation as:
4[(x² - 6x + 9) - 9] + 4[(y² - 8y + 16) - 16] = 96
This simplifies to:
4[(x - 3)² - 9] + 4[(y - 4)² - 16] = 96
Further simplifying gives:
4(x - 3)² - 36 + 4(y - 4)² - 64 = 96
Combine like terms:
4(x - 3)² + 4(y - 4)² = 196
Divide through by 4 to get the equation in standard form:
(x - 3)² + (y - 4)² = 49
So, the center of the circle is at (3,4) and the radius is √49 = 7.
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