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 $y = k \pm \sqrt{r^2 - h^2}$.
Answer for screen readers
The x-intercepts of the circle are given by:
$$ x = h \pm \sqrt{r^2 - k^2} $$
And the y-intercepts are given by:
$$ y = k \pm \sqrt{r^2 - h^2} $$
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.