To classify how a line meets a curve, set their equations equal, rearrange to a quadratic equal to zero, and look at b² − 4ac. Positive means two points, zero means the line is a tangent, and negative means no intersection.
This skill sits in quadratic structure and discriminants and returns when you solve simultaneous equations and problems with a parameter.
Why does the discriminant decide the number of intersections?
The x-coordinates where a line and curve meet are the solutions of “line = curve”. That is a quadratic in x. The quadratic formula has a square root of b² − 4ac in it, so the sign of that number decides whether the square root exists and whether the two values collapse into one.
You never need to find the points to classify the situation. That saves time in questions that only ask “how many times” or “for what values of k”.
How do I set the question up?
- Equate the line and the curve.
- Bring every term to one side so the equation reads ax² + bx + c = 0.
- Write down a, b and c with their signs, including any letter such as k.
- Form b² − 4ac and simplify.
- Apply the rule: more than 0 for two points, equal to 0 for a tangent, less than 0 for none.
Worked example
The curve is y = x² − 4x + 5 and the line is y = 2x + k. Find the values of k for which the line meets the curve at two distinct points, and find the point of contact when it is a tangent.
Step 1, equate: x² − 4x + 5 = 2x + k.
Step 2, rearrange: x² − 6x + (5 − k) = 0.
Step 3, identify: a = 1, b = −6, c = 5 − k.
Step 4, discriminant: (−6)² − 4(1)(5 − k) = 36 − 20 + 4k = 16 + 4k.
Step 5, apply: two points when 16 + 4k > 0, which gives k > −4. Tangent when k = −4. No intersection when k < −4.
Point of contact at k = −4: x² − 6x + 9 = 0, so (x − 3)² = 0 and x = 3. Then y = 2(3) − 4 = 2.
Check: the curve at x = 3 gives 9 − 12 + 5 = 2. Both agree, so the contact point is (3, 2).
The mistake to watch for
A typical slip is to read a, b and c before moving the terms across.
Mistaken working: x² − 4x + 5 = 2x + k, so a = 1, b = −4, c = 5 − k, giving 16 − 4(5 − k) = 4k − 4. Tangent when k = 1.
The 2x on the right was never combined with −4x, so b is wrong.
Test the answer: at k = 1 the equation is x² − 6x + 4 = 0, which has discriminant 36 − 16 = 20, not 0. The correction is to simplify fully to ”= 0” before reading any coefficient.
Check yourself
1. Does the line y = x − 2 meet the curve y = x² + 3x + 1?
Show answer
x² + 3x + 1 = x − 2 gives x² + 2x + 3 = 0. The discriminant is 2² − 4(1)(3) = 4 − 12 = −8. It is negative, so the line does not meet the curve.
2. How many times does y = 3x² − 2x − 1 cross the x-axis?
Show answer
Set y = 0. The discriminant is (−2)² − 4(3)(−1) = 4 + 12 = 16, which is positive. So it crosses twice. Check: the roots are (2 ± 4)/6 = 1 and −1/3.
3. The line y = 2x + k meets the curve y = x² − 2x + 6. Find k when the line is a tangent, and the point of contact.
Show answer
x² − 2x + 6 = 2x + k gives x² − 4x + (6 − k) = 0. The discriminant is 16 − 4(6 − k) = 4k − 8. Tangent when k = 2. Then x² − 4x + 4 = 0, so x = 2 and y = 2(2) + 2 = 6. Contact point (2, 6). Check: the curve at x = 2 gives 4 − 4 + 6 = 6.
Where this leads next
Try solve a parameter condition for repeated roots, which applies the same idea to an equation with a letter in it. The quadratic structure explorer shows the discriminant and the graph together, and the non-calculator working trainer helps keep your arithmetic exact.
If the setup step is where your marks go, our teachers can work through your own questions in online one-to-one Additional Mathematics tuition.