In terms of pi, what is the area of the circle defined by the equation $2x^2+2y^2+10x-6y-18=0?
+2666 Isolate the constant: \(2x^2 + 2y^2 + 10x - 6y = 18\)
Divide the entire equation by 2: \(x^2 + y^2 + 5x - 3y = 9\)
Add \(8.5\) to both sides so we can complete the square for both x and y: \(x^2 + y^2 + 5x - 3y +8.5= 17.5\)
Complete the square: \((x+2.5)^2 + (y - 1.5)^2 = 17.5\)
Note the constant on the left-hand side equals the radius of the circle squared, meaning the area of the circle is \(\color{brown}\boxed{17.5 \pi}\)
