Question

review for final examination
score: 36/50 answered: 36/50
question 37
write the equation of the circle centered at (7, -6) that passes through (9, 4) .
question help: video 1 video 2

Explanation:

Step1: Recall the standard form of a circle's equation

The standard form of a circle's equation is \((x - h)^2+(y - k)^2=r^2\), where \((h,k)\) is the center of the circle. Given the center \((h,k)=(7,-6)\), the equation becomes \((x - 7)^2+(y+6)^2=r^2\).

Step2: Calculate the radius \(r\)

The radius \(r\) is the distance between the center \((7,-6)\) and the point \((9,4)\) on the circle. Use the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\). Here, \(x_1 = 7,y_1=-6,x_2 = 9,y_2 = 4\). Then \(r=\sqrt{(9 - 7)^2+(4+6)^2}=\sqrt{4 + 100}=\sqrt{104}\). So \(r^2=104\).

Step3: Write the final equation of the circle

Substitute \(r^2 = 104\) into the equation \((x - 7)^2+(y + 6)^2=r^2\). The equation of the circle is \((x - 7)^2+(y+6)^2=104\).

Answer:

\((x - 7)^2+(y + 6)^2=104\)