How to find intercepts of a circle?

Understand the Problem

The question is asking how to determine the intercepts of a circle, specifically the points where the circle intersects the axes or any other line. To solve it, we need to use the standard equation of a circle and possibly substitute values to find the x and y intercepts.

Answer

The x-intercepts are $x = h \pm \sqrt{r^2 - k^2}$ and the y-intercepts are $y = k \pm \sqrt{r^2 - h^2}$.

Steps to Solve

Identify the standard equation of the circle

The standard equation of a circle with center at $(h, k)$ and radius $r$ is given by:

$$ (x - h)^2 + (y - k)^2 = r^2 $$

Finding the x-intercepts

To find the x-intercepts, set $y = 0$ in the circle's equation and solve for $x$:

$$ (x - h)^2 + (0 - k)^2 = r^2 $$

This simplifies to:

$$ (x - h)^2 + k^2 = r^2 $$

Now, we can solve for $x$:

$$ (x - h)^2 = r^2 - k^2 $$

Taking the square root of both sides:

$$ x - h = \pm\sqrt{r^2 - k^2} $$

Thus, the x-intercepts are:

$$ x = h \pm \sqrt{r^2 - k^2} $$

Finding the y-intercepts

To find the y-intercepts, set $x = 0$ in the circle's equation and solve for $y$:

$$ (0 - h)^2 + (y - k)^2 = r^2 $$

This simplifies to:

$$ h^2 + (y - k)^2 = r^2 $$

Now, we can solve for $y$:

$$ (y - k)^2 = r^2 - h^2 $$

Taking the square root of both sides:

$$ y - k = \pm\sqrt{r^2 - h^2} $$

Thus, the y-intercepts are:

$$ y = k \pm \sqrt{r^2 - h^2} $$

The x-intercepts of the circle are given by:

$$ x = h \pm \sqrt{r^2 - k^2} $$

And the y-intercepts are given by:

$$ y = k \pm \sqrt{r^2 - h^2} $$

More Information

The intercepts of a circle provide important points that can help in graphing the equation. They also allow you to understand how the circle intersects the coordinate axes.

Tips

Forgetting to set $y = 0$ when finding x-intercepts or $x = 0$ when finding y-intercepts.
Not taking into account the radius and the position of the center which can affect whether there are real intercepts.

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