Skill: Solve coordinate geometry problems involving lines and circles

To find where a line and a circle intersect, use substitution:

1. Write the line in a form such as $y=mx+b$ or identify its given value.
2. Substitute the line equation into the circle equation.
3. Solve the resulting equation.
4. Substitute the solution back into the line to find the corresponding coordinates.
5. State the intersection point or points.

Example

Find the points where the line $y=3$ intersects the circle

$$x^2+y^2=25.$$

The circle has center $(0,0)$ and radius $5$. Since the line is $y=3$, substitute $3$ for $y$ in the circle equation:

$$x^2+3^2=25.$$

Simplify:

$$x^2+9=25$$ $$x^2=16.$$

Taking the square root gives two possible values:

$$x=4 \quad \text{or} \quad x=-4.$$

The line equation tells us that $y=3$ for both points. Therefore, the intersection points are

$$\boxed{(4,3)\text{ and }(-4,3)}.$$

There are two intersection points because the line passes through the circle at two locations. In general, solving the substituted equation tells you the number of intersections: two real solutions mean two points, one real solution means the line is tangent to the circle, and no real solutions mean the line does not meet the circle.