A circle is defined by the equation \((x - 5)^2 + (y - 7)^2 = 81\). Find the expanded form of this circle.
Answer
Finally, simplify the equation by subtracting 81 from both sides to get \(x^2 - 10x + y^2 - 14y - 7 = 0\). This is the expanded form of the circle.
Step 1 ：The equation of a circle is given by \((x - h)^2 + (y - k)^2 = r^2\), where \((h, k)\) are the coordinates of the center of the circle and \(r\) is the radius.
Step 2 ：Substitute \(h = 5\), \(k = 7\), and \(r = 9\) into the equation, we get \((x - 5)^2 + (y - 7)^2 = 81\).
Step 3 ：Now, expand the squared terms. So, \((x - 5)^2\) becomes \(x^2 - 10x + 25\), and \((y - 7)^2\) becomes \(y^2 - 14y + 49\).
Step 4 ：Then, substitute these expressions back into the equation, we get \(x^2 - 10x + 25 + y^2 - 14y + 49 = 81\).
Step 5 ：Finally, simplify the equation by subtracting 81 from both sides to get \(x^2 - 10x + y^2 - 14y - 7 = 0\). This is the expanded form of the circle.
