67 total
By the end of these notes, you will be able to:
When you draw a straight line near or through a circle, one of three things can happen:
| Situation | What it looks like | Number of intersection points |
|---|---|---|
| Line misses the circle | Line is completely outside | 0 points |
| Line is a tangent | Line just grazes (touches) the edge | 1 point |
| Line is a chord | Line cuts through and comes out the other side | 2 points |
💡 Think of it this way: Imagine rolling a ball (the circle) along the ground (the line). If the ball just barely touches the ground, that's a tangent. If you could somehow push the line through the ball, it would enter and exit — that's a chord.
To find out where (and whether) a line and circle intersect, you use simultaneous equations — this means you solve both equations at the same time.
Here's the step-by-step process:
Step 1: Rearrange the equation of the straight line to make either x or y the subject (i.e., get x=... or y=...).
Step 2: Substitute (replace) this expression into the equation of the circle.
Step 3: Expand and simplify. You will always end up with a quadratic equation — an equation in the form ax2+bx+c=0.
Step 4: Solve the quadratic. The number of solutions tells you what type of intersection you have.
Once you have your quadratic equation ax2+bx+c=0, you use something called the discriminant to figure out the type of intersection — without even fully solving the equation!
📌 The discriminant is the expression b2−4ac. It's calculated from the numbers in your quadratic equation.
Here is what the discriminant tells you:
| Value of b2−4ac | Type of roots | What this means for the line and circle |
|---|---|---|
| b2−4ac>0 | Two different (distinct) real roots | Line intersects circle at two points → line is a chord |
| b2−4ac=0 | One repeated real root | Line touches circle at one point → line is a tangent |
| b2−4ac<0 | No real roots | Line does not meet the circle at all |
💡 Plain English: The discriminant is like a test. A positive result = two crossing points. Zero = just touching. Negative = no meeting at all.
Sign in to view full notes