Question - Finding the Diameter of a Circle from a Given Equation
Solution:
Complete the square for $$ x $$ and $$ y $$:$$ x^2 + 22x = x^2 + 22x + (22/2)^2 - (22/2)^2 = (x+11)^2 - 121 $$$$ y^2 + 30y = y^2 + 30y + (30/2)^2 - (30/2)^2 = (y+15)^2 - 225 $$Rewrite the equation:$$ (x+11)^2 - 121 + (y+15)^2 - 225 + 90 = 0 $$$$ (x+11)^2 + (y+15)^2 = 121 + 225 - 90 $$$$ (x+11)^2 + (y+15)^2 = 256 $$The equation $$ (x+11)^2 + (y+15)^2 = 256 $$ represents a circle with radius $$ r^2 = 256 $$, so $$ r = 16 $$.Diameter $$ d = 2r = 2 \times 16 = 32 $$. The diameter of the circle is 32.
Tutor Solution:
To solve for the diameter of the circle given by the equation $$x^2 + y^2 - 22x + 30y + 90 = 0$$, we need to put the equation in the standard form of a circle's equation, which is $$(x - h)^2 + (y - k)^2 = r^2$$, where $$(h, k)$$ is the center of the circle and $$r$$ is the radius.First, we need to complete the square for both the $$x$$ and $$y$$ terms.For the $$x$$ terms: We have $$x^2 - 22x$$. To complete the square, we need to add and subtract the square of half the coefficient of $$x$$, which is $$\left(\frac{-22}{2}\right)^2 = 11^2 = 121$$. Add 121 and subtract 121 next to the $$x$$ terms.$$x^2 - 22x + 121 - 121$$For the $$y$$ terms: We have $$y^2 + 30y$$. Similarly, we will add and subtract the square of half the coefficient of $$y$$, which is $$\left(\frac{30}{2}\right)^2 = 15^2 = 225$$. Add 225 and subtract 225 next to the $$y$$ terms.$$y^2 + 30y + 225 - 225$$Now the equation will look like this:$$x^2 - 22x + 121 + y^2 + 30y + 225 - 121 - 225 + 90 = 0$$Group the terms to form perfect squares for $$x$$ and $$y$$, and move the constants to the other side:$$(x^2 - 22x + 121) + (y^2 + 30y + 225) = 121 + 225 - 90$$Now we rewrite the groups as squares of binomials:$$(x - 11)^2 + (y + 15)^2 = 256$$We now have the standard form:$$(x - 11)^2 + (y + 15)^2 = 16^2$$Where the radius $$r$$ of the circle is 16 (since $$r^2 = 256 \Rightarrow r = 16$$).The diameter $$d$$ of a circle is twice the radius:$$d = 2r$$Substitute $$r = 16$$ into the equation for $$d$$:$$d = 2 \times 16 = 32$$So, the diameter of the circle is 32 units.
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