Write the standard form of the equation of the circle with the given characteristics.
Center: (6, 1); Solution point: (5, 9)

A) $( x + 6 ) ^ { 2 } + ( y + 1 ) ^ { 2 } = 65$
B) $( x - 6 ) ^ { 2 } + ( y + 1 ) ^ { 2 } - 65 = 0$
C) $( x + 6 ) ^ { 2 } + ( y + 1 ) ^ { 2 } + 65 = 0$
D) $( x - 6 ) ^ { 2 } + ( y - 1 ) ^ { 2 } = 65$
E) $( x - 1 ) ^ { 2 } + ( y - 6 ) ^ { 2 } = 65$

Question

Write the standard form of the equation of the circle with the given characteristics.  
Center: (6, 1); Solution point: (5, 9)
 
A) $( x + 6 ) ^ { 2 } + ( y + 1 ) ^ { 2 } = 65$
B) $( x - 6 ) ^ { 2 } + ( y + 1 ) ^ { 2 } - 65 = 0$
C) $( x + 6 ) ^ { 2 } + ( y + 1 ) ^ { 2 } + 65 = 0$
D) $( x - 6 ) ^ { 2 } + ( y - 1 ) ^ { 2 } = 65$
E) $( x - 1 ) ^ { 2 } + ( y - 6 ) ^ { 2 } = 65$

Quizplus · Accepted Answer

The standard form of a circle's equation is $(x - h)^2 + (y - k)^2 = r^2$, where $(h, k)$ is the center and $r$ is the radius. The center is given as (6, 1), so $h = 6$ and $k = 1$, which matches with option D. To find $r^2$, calculate the distance between the center (6, 1) and the solution point (5, 9) using the distance formula: $\sqrt{(5-6)^2 + (9-1)^2} = \sqrt{1^2 + 8^2} = \sqrt{65}$. Thus, $r^2 = 65$, confirming option D is correct.

Write the Standard Form of the Equation of the Circle (x+6)2+(y+1)2=65( x + 6 ) ^ { 2 } + ( y + 1 ) ^ { 2 } = 65(x+6)2+(y+1)2=65

Write the Standard Form of the Equation of the Circle $( x + 6 ) ^ { 2 } + ( y + 1 ) ^ { 2 } = 65$