Lines and circles
Objective. Find intersections and test tangency algebraically.
- 1. Learn
- 2. Worked example
- 3. Practice
- 4. Feedback
- 5. Continue
Learn: the key idea
To find where a line meets a circle, substitute the line's equation into the circle's equation to form a single quadratic, then examine its discriminant: two solutions mean the line is a secant crossing at two points, one repeated solution means the line is a tangent, and no real solutions mean the line misses the circle entirely. For the circle x²+y²=25 and the line y=x+1, substituting gives x²+(x+1)²=25, leading to 2x²+2x−24=0, which simplifies to x²+x−12=0 with roots x=3 and x=−4, so the line is a secant. Testing tangency this way avoids relying purely on a diagram.
Worked example
Show that the line y=x+5 is tangent to the circle x²+y²=12.5 by examining intersections.
- 1Substitute y=x+5 into the circle: x²+(x+5)²=12.5
- 2Expand and simplify: 2x²+10x+25=12.5, so 2x²+10x+12.5=0
- 3Divide by 2: x²+5x+6.25=0, giving discriminant 25−25=0
- 4A zero discriminant confirms exactly one intersection, so the line is tangent
Answer: Tangent, since the discriminant equals 0
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Start practisingCommon mistake. Students assume a line touching a circle on a rough sketch must be a tangent without checking the discriminant algebraically; always verify tangency by confirming discriminant equals exactly zero.
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