A coordinate geometry problem often gives you two descriptions of the same point. A point on a circle must satisfy its equation; a point on a line must satisfy the line's equation. An intersection must satisfy both at once. That makes finding an intersection a systems-of-equations problem, even when one of the graphs is curved.
We will connect substitution, quadratic factoring, and the distance formula. The aim is to explain why an answer works as well as calculate it.
The problem: find every shared point
Find all real intersection points of the circle x2 + y2 = 25 and the line y = x + 1. Then find the length of the chord joining them. Coordinates are measured in units, so a length will be in units.
The circle has center (0, 0) and radius 5 because a circle centered at the origin has equation x2 + y2 = r2. The line has slope 1 and y-intercept 1. A sketch suggests two crossings, but the equations will give their exact coordinates.
Step 1: substitute the line into the circle
Since every point on the line has y = x + 1, replace y in the circle's equation with the entire expression x + 1. Keep parentheses around it when squaring:
x² + (x + 1)² = 25
x² + x² + 2x + 1 = 25
2x² + 2x − 24 = 0
x² + x − 12 = 0The middle term 2x matters: (x + 1)2 means (x + 1)(x + 1), which is x2 + 2x + 1. Expanding it as x2 + 1 would produce different, incorrect intersection points.
Step 2: factor, then recover both coordinates
We need two numbers whose product is −12 and whose sum is 1. They are 4 and −3:
(x + 4)(x − 3) = 0
x = −4 or x = 3Each x-value needs its matching y-value from the original line. If x = −4, then y = −3. If x = 3, then y = 4. The intersection points are (−4, −3) and (3, 4). Do not mix the x-value from one solution with the y-value from the other.
Step 3: check each point in both equations
- (−4, −3): 16 + 9 = 25, and −3 = −4 + 1.
- (3, 4): 9 + 16 = 25, and 4 = 3 + 1.
Both points pass both checks. Negative coordinates are valid here: they describe a position relative to the origin. The radius and the chord length must be nonnegative, but individual coordinates need not be.
Step 4: use the points to find a geometric length
A chord is a segment whose endpoints lie on a circle. The horizontal and vertical changes between our endpoints are both 7. Apply the distance formula, which comes from the Pythagorean theorem:
d = √((3 − (−4))² + (4 − (−3))²)
= √(7² + 7²)
= √98 = 7√2 unitsThe exact length is 7√2 units, approximately 9.90 units. The diameter is 10 units, so this chord is slightly shorter than the diameter, as the picture suggests. A value larger than 10 would tell us to recheck our work.
Why a line can meet a circle zero, one, or two times
Keep the same circle and consider horizontal lines. For y = 3, substitution gives x2 = 16: there are two intersection points, (−4, 3) and (4, 3). For y = 5, it gives x2 = 0: there is one distinct point, (0, 5), where the line is tangent. For y = 6, it gives x2 = −11, which has no real solution, so the graphs do not meet.
The number of distinct real algebraic solutions matches the number of distinct geometric intersections. This gives meaning to solving the quadratic rather than treating its roots as an isolated calculation.
Try the method yourself
Find every intersection of x2 + y2 = 25 with y = x − 1. Check your points in both equations before opening the answer.
Show the worked answer
Substitute to get x2 + (x − 1)2 = 25, so x2 − x − 12 = 0. Factoring gives (x − 4)(x + 3) = 0. The points are (4, 3) and (−3, −4). Both have squared coordinates summing to 25, and both satisfy y = x − 1.
Continue with the Algebra 2 Honors study guide for systems and quadratics, or the Geometry Honors study guide for circles and coordinate reasoning.