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In terms of pi, what is the area of the circle defined by the equation 2x^2+2y^2+10x-6y-48=0

Aug 2, 2022

#1
+2275
+2

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 $$\color{brown}\boxed{32.5 \pi}$$

Aug 2, 2022

#1
+2275
+2

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 $$\color{brown}\boxed{32.5 \pi}$$

BuilderBoi Aug 2, 2022