When a line and a circle intersect, their common points must satisfy both equations. Substitution turns the geometry into a quadratic equation whose roots represent the possible intersection coordinates.
1. Substitute the line into the circle
Find the intersections of x²+y²=25 and y=x+1.
Substitute y=x+1 into the circle:
x²+(x+1)²=25.
This gives 2x²+2x−24=0, or x²+x−12=0=(x+4)(x−3).
Thus x=−4 or 3. Corresponding y-values are −3 and 4. The intersections are (−4,−3) and (3,4).
2. Why a quadratic appears
A line equation is first degree, while the circle contains squared coordinates. Substituting the linear relation into the circle generally produces a quadratic in one variable.
3. Discriminant counts intersections
- Δ>0: two distinct real intersection points.
- Δ=0: one repeated intersection, so the line is tangent.
- Δ<0: no real intersection.
This is the same discriminant logic used for quadratic roots, now interpreted geometrically.
4. Worked tangency condition
Determine whether y=4 intersects x²+y²=9.
Substitute y=4: x²+16=9, so x²=−7. There is no real solution, so the line lies outside the circle and does not intersect it.
5. A secant line
For y=0 and x²+y²=16, x²=16, so x=±4. The horizontal line cuts the circle at two points. It is a secant through the diameter.
6. A tangent line
For y=3 and x²+y²=9, x²+9=9, so x=0 is a repeated solution. The tangent point is (0,3).
7. Unknown line parameter
Find k such that y=k is tangent to x²+y²=25.
Substitute: x²+k²=25. Tangency requires exactly one real x-value, so x=0 and k²=25. Therefore k=5 or k=−5.
Geometrically these are the top and bottom horizontal tangents.
8. Tangency through discriminant
For a line y=mx+c intersecting x²+y²=r², substitution gives a quadratic in x. Setting its discriminant to zero provides a condition linking m, c and r.
This is especially useful when the tangent point is not supplied.
9. Check both coordinates
After solving for x, substitute into the line to find y, then verify the pair in the circle if needed. A root of the substituted quadratic should represent a common point, not just a partial coordinate.
10. Common mistakes
- Solving the quadratic for x but forgetting to find y.
- Calling one algebraic root a tangent without checking multiplicity.
- Setting Δ=0 before forming the correct intersection quadratic.
- Assuming every line through the circle centre is tangent; it is usually a secant.
11. Practice
- Find the intersections of x²+y²=13 and y=x+1.
- Determine whether y=6 meets x²+y²=25.
- Find k if y=k is tangent to x²+y²=36.
- Explain the geometric meaning of Δ<0 after substituting a line into a circle.
- Find the intersections of x²+y²=10 and y=3x.
12. Answers
- x²+(x+1)²=13 gives x²+x−6=0, so x=2 or −3; points (2,3) and (−3,−2).
- No; x²+36=25 has no real solution.
- k=±6.
- The line and circle have no real common point.
- x²+9x²=10 gives x=±1; points (1,3) and (−1,−3).
Continue with Finding Tangents to Circles, Discriminants in Additional Mathematics, or return to the Additional Mathematics Hub.