In terms of pi, what is the area of the circle defined by the equation 2x^2+2y^2+10x-6y-48=0
+2666 Simplify: \(x^2 + y^2 + 5x - 3y - 24 = 0 \)
Add \(1.5^2 \) and \(2.5^2\) to both sides to complete the square: \(x^2 + y^2 + 5x - 3y - 24 + 1.5^2 + 2.5^2 = 1.5^2 + 2.5^2\)
Now, add 24 to both sides: \(x^2 + y^2 + 5x - 3y + 1.5^2 + 2.5^2 = 24 + 1.5^2 + 2.5^2\)
Complete the square on x: \((x+2.5)^2 + y^2 - 3y - 24 = 1.5^2 + 2.5^2\)
Complete the square on y: \((x+2.5)^2 + (y-1.5)^2 = 1.5^2 +2.5^2 + 24\)
Simplify the right-hand side: \((x+2.5)^2 + (y - 1.5)^2 = 32.5\)
The radius of this circle is \(\sqrt{32.5}\), so the area is \(\color{brown}\boxed{32.5 \pi}\)
+2666 Simplify: \(x^2 + y^2 + 5x - 3y - 24 = 0 \)
Add \(1.5^2 \) and \(2.5^2\) to both sides to complete the square: \(x^2 + y^2 + 5x - 3y - 24 + 1.5^2 + 2.5^2 = 1.5^2 + 2.5^2\)
Now, add 24 to both sides: \(x^2 + y^2 + 5x - 3y + 1.5^2 + 2.5^2 = 24 + 1.5^2 + 2.5^2\)
Complete the square on x: \((x+2.5)^2 + y^2 - 3y - 24 = 1.5^2 + 2.5^2\)
Complete the square on y: \((x+2.5)^2 + (y-1.5)^2 = 1.5^2 +2.5^2 + 24\)
Simplify the right-hand side: \((x+2.5)^2 + (y - 1.5)^2 = 32.5\)
The radius of this circle is \(\sqrt{32.5}\), so the area is \(\color{brown}\boxed{32.5 \pi}\)
