Q:

# The general form of the equation of a circle is 7x2 + 7y2 − 28x + 42y − 35 = 0The equation of this circle in standard form isA. (x - 2)^2 + (y + 3)^2 = 5 B. (x - 2)^2 + (y + 3)^2 = 18 C. (x - 4)^2 + (y + 6)^2 = 24 D. (x + 4)^2 + (y - 6)^2 = 24 E. (x + 2)^2 + (y - 3)^2 = 35 The center of the circle is at the point __ A. (-14,21) B. (-2,3) C. (2,-3) D. (-4,6) E. (4,-6) And it's radius is ___ units. A. 5 1/2 B. 3(2 1/2) C. 2(6 1/2) D. 35 1/2

Accepted Solution

A:
Greetings!Answer:The standard form equation is B.The centre of the circle is BThe radius is B.Step-by-step explanation:Firstly, the equation can all be divided by 7:(7x² + 7y²− 28x + 42y − 35)  ÷ 7 =x² + y² - 4x + 6y - 5 = 0x² - 4x + y² + 6y - 5 = 0Now add $$\frac{1}{2}* b$$0.5 * 4 = 2, 2² = 4 so 40.5 * 6 = 3. 3² = 9 so 9(x² - 4x + 4) + (y² + 6y + 9) - 4 - 9 - 5 = 0As the standard form of a circle is (x - a)² + (y - b)²= r² , we need to change this to this. Simply find a number that multiplies to 4 and adds to -4 ( -2 ) and a number that multitplies to 9 and adds to 6 (3):(x - 2)² + (y + 3)²Also, we need to subtract the square of the number inside the brackets, as this is completing the square:-2 * -2 = 43 * 3 = 9So we need to subtract 4 and 9, and 5 as that was in the original:(x - 2)² + (y + 3)² - 4 - 9 - 5 = 0To find the radius, we need to move this numbers over to the other side making them positives:(x - 2)² + (y + 3)² = 4 + 9 + 5(x - 2)² + (y + 3)² = 18So the equation is B.The centre of the circle in (x - a)² + (y - b)² is $$a$$ and $$b$$So in this equation the centre is -2, 3. Which is B.The radius is the square root of whatever (x - a)² + (y - b)² equals, so the square root of 18 is the radius:√18 = √9√2 = 3√2= $$3 * (2x^{\frac{1}{2} } )$$ So your answer is B.
Hope this helps!