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A circle in the $xy$ -plane has the equation shown. If the $x$ -coordinate of a point on the circle is $-3$ , what is a possible corresponding $y$ -coordinate?

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Rational functions: asymptotes and excluded values

Full solution

Q. A circle in the

xy

-plane has the equation shown. If the

x

-coordinate of a point on the circle is

-3

, what is a possible corresponding

y

-coordinate?

Circle Equation Assumption: We need the equation of the circle to find the $y$ -coordinate. Let's assume the equation of the circle is $(x - h)^2 + (y - k)^2 = r^2$ , where $(h, k)$ is the center and $r$ is the radius. But we don't have the actual equation here, so we'll use a generic one: $x^2 + y^2 = r^2$ .
Plug in $x = -3$ : Plug in $x = -3$ into the equation $x^2 + y^2 = r^2$ to find $y$ .\(\newline\)( $-3$ )^ $2$ + $y^2 = r^2$ \(\newline\) $9 + y^2 = r^2$
Solve for y: We need to solve for y. Subtract $9$ from both sides to isolate $y^2$ . $\newline$ $y^2 = r^2 - 9$
Take Square Root: To find $y$ , we take the square root of both sides. Remember, there are two solutions because a square root has both a positive and negative solution. $y = \pm\sqrt{(r^2 - 9)}$
Error: Missing Circle Equation: Oops, we made a mistake. We don't know the value of $r$ , the radius of the circle. Without the actual equation of the circle, we can't find the exact $y$ -coordinate. We need the specific equation to proceed.

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