Answer:
(x +4)² +(y -5)² = 100
Step-by-step explanation:
Given circle A has center X(-4, 5) and passes through Y(2, -3), you want its equation.
Circle B
The equation of circle B with center Y is given as ...
x² +y² -4x +6y -12 = 0
This can be rearranged to standard form by completing the square for x and for y.
(x² -4x +4) +(y² +6y +9) = 12 +4 +9
(x -2)² +(y +3)² = 25
This tells us the center of circle B is Y(2, -3).
Circle A
The standard form equation for circle A with X(-4, 5) as its center will be ...
(x +4)² +(y -5)² = r²
where r is the distance XY. Using the Pythagorean theorem or the distance formula, we have ...
r² = (2 -(-4))² +(-3 -5)² = 6² +(-8)² = 100
Then the standard form equation for circle A is ...
(x +4)² +(y -5)² = 100 . . . . . . circle A
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Additional comment
The standard form equation for a circle with center (h, k) and radius r is ...
(x -h)² +(y -k)² = r²
The general form equation can be found by expanding this and collecting constants on the left side.
x² +y² +8x -10y -59 = 0 . . . . . . circle A
