Question

question 5 of 10 which of the following circles have their centers in the second quadrant? check all that apply. a. $(x + 12)^2+(y - 9)^2 = 19$ b. $(x + 3)^2+(y - 2)^2 = 8$ c. $(x - 2)^2+(y + 7)^2 = 64$ d. $(x - 5)^2+(y + 5)^2 = 9$

Explanation:

Step1: Recall the standard form of a circle equation

The standard form of a circle equation is \((x - h)^2+(y - k)^2=r^2\), where \((h,k)\) is the center of the circle.

Step2: Find the centers of each circle

• For circle A: \((x + 12)^2+(y - 9)^2=19\), we have \(h=- 12\), \(k = 9\). So the center is \((-12,9)\).
• For circle B: \((x + 3)^2+(y - 2)^2=8\), we have \(h=-3\), \(k = 2\). So the center is \((-3,2)\).
• For circle C: \((x - 2)^2+(y + 7)^2=64\), we have \(h = 2\), \(k=-7\). So the center is \((2,-7)\).
• For circle D: \((x - 5)^2+(y + 5)^2=9\), we have \(h = 5\), \(k=-5\). So the center is \((5,-5)\).

Step3: Determine which centers are in the second quadrant

In the second quadrant, \(x<0\) and \(y>0\).

• For center \((-12,9)\) (circle A): \(-12<0\) and \(9>0\).
• For center \((-3,2)\) (circle B): \(-3<0\) and \(2>0\).
• For center \((2,-7)\) (circle C): \(2>0\) (not in the second quadrant).
• For center \((5,-5)\) (circle D): \(5>0\) (not in the second quadrant).

Answer:

A. \((x + 12)^2+(y - 9)^2=19\), B. \((x + 3)^2+(y - 2)^2=8\)