Final answer:
The standard form of the equation for the circle with center (10,-5) and passing through (8,7) is calculated using the distance formula to find the radius and then plugging the center and radius into the standard circle equation, resulting in (x-10)² + (y+5)² = 148.
Step-by-step explanation:
The question asks to find the standard form of the equation for a circle with a given center at (10,-5) and that passes through the point (8,7). To find the equation, we require the center coordinates and the radius of the circle. The radius can be found by calculating the distance between the center and the point on the circle using the distance formula: √((x2-x1)² + (y2-y1)²). Here, (x1, y1) is the center of the circle, and (x2, y2) is the point it passes through.
Substituting the values we get: radius = √((8-10)² + (7-(-5))²) = √((-2)² + (12)²) = √(4 + 144) = √148. The standard form of the equation of a circle is (x-h)² + (y-k)² = r², where (h, k) is the center and r is the radius. Therefore, substituting the values of the center and radius into this equation, we get (x-10)² + (y+5)² = 148. This is the standard form equation of the circle for the given conditions.