When a Line Meets, Touches or Misses a Curve

Substitute a line into a curve, then use the discriminant to decide if the line cuts it twice, touches it or misses it, and find unknown constants.

In the exam for Edexcel Further Pure Maths 4PM1, IGCSE Additional Maths 0606 · 4 real past-paper questions below

اقرأ هذا الدرس مشروحاً بالعربي ←

Cut, touch or miss

Picture a straight road running near a curved hill. The road might cut through the hill, just touch its edge, or pass by without meeting it. A line and a curve have the same three choices.

A meeting point lies on both graphs, so its x and y fit both equations. Put the line into the curve and you get one equation in x. Each real root of it is one meeting point.

Rule

$b^{2} - 4 a c > 0$ : two distinct roots. The line cuts (intersects) the curve at two points.
$b^{2} - 4 a c = 0$ : two equal roots. The line touches the curve at one point: it is a tangent.
$b^{2} - 4 a c < 0$ : no real roots. The line does not meet the curve.
“The line meets the curve” means at least one point: $b^{2} - 4 a c \geq 0$ .

Worked example

1
Show that the line $y = 2 x + 1$ is a tangent to the curve $y = x^{2} - 2 x + 5$ .
2
$x^{2} - 2 x + 5 = 2 x + 1$ — At a meeting point both y values are equal.
3
$x^{2} - 4 x + 4 = 0$ — Move everything to one side so it equals 0.
4
$(- 4)^{2} - 4 (1) (4) = 0$ — Discriminant with a = 1, b = −4, c = 4.
5
$b^{2} - 4 a c = 0$ — Equal roots, so the line touches the curve: it is a tangent.
6
$x = 2, y = 5$ — $(x - 2)^{2} = 0$ gives $x = 2$ . Then $y = 2 (2) + 1 = 5$ . It touches at (2, 5).

Key idea

The word tangent comes from the Latin word tangere, which means “to touch”. A tangent is the line that only touches.

Tangent: find k

Now the line has an unknown, like $y = k x - 2$ . The question asks for the values of k that make it a tangent. Use the same method, then set the discriminant to 0.

Steps

1
1. Substitute the line into the curve.
2. Move every term to one side so it equals 0. Group the x terms: the bracket in front of x is b.
3. Write the condition: $b^{2} - 4 a c = 0$ for a tangent.
4. Solve for k. Then, if asked, find the point where it touches.

Worked example

1
The line $y = k x - 2$ is a tangent to the curve $y = x^{2} + 2 x + 2$ . Find the possible values of k.
2
$x^{2} + 2 x + 2 = k x - 2$ — Substitute the line.
3
$x^{2} + (2 - k) x + 4 = 0$ — Everything on one side. The x terms group as $(2 - k) x$ .
4
$(2 - k)^{2} - 4 (1) (4) = 0$ — Tangent: discriminant equals 0.
5
$(2 - k)^{2} = 16$ — Add 16 to both sides.
6
$2 - k = \pm 4$ — Square root, and keep both signs.
7
$k = - 2 or k = 6$ — From $2 - k = 4$ and $2 - k = - 4$ .

Worked example

1
Find the point where the tangent with $k = 6$ touches the curve.
2
$x^{2} - 4 x + 4 = 0$ — Put $k = 6$ into $x^{2} + (2 - k) x + 4 = 0$ .
3
$(x - 2)^{2} = 0 \Rightarrow x = 2$ — A repeated root, as a tangent must give.
4
$y = 6 (2) - 2 = 10$ — Use the line to find y. The point is (2, 10).

Common mistake

Using a, b and c from the curve alone. The discriminant must come from the new equation, after you substitute and collect everything on one side.

Common mistake

Forgetting the minus root. $(2 - k)^{2} = 16$ gives $2 - k = 4$ or $2 - k = - 4$ , so there are two tangents.

Key idea

Substitute, make it equal 0, then use the discriminant. Every question in this lesson starts with these three moves.

Two points or none

Sometimes the line must cut the curve at two points, or miss it completely. Then the condition is an inequality, and the answer is a range of values.

Worked example

1
Find the set of values of k for which the line $y = k x + 1$ meets the curve $y = x^{2} + 3 x + 3$ at two distinct points.
2
$x^{2} + (3 - k) x + 2 = 0$ — Substitute and move everything to one side.
3
$(3 - k)^{2} - 4 (1) (2) > 0$ — Two distinct points: discriminant greater than 0.
4
$k^{2} - 6 k + 1 > 0$ — Expand: $9 - 6 k + k^{2} - 8$ .
5
$k = \frac{6 \pm 32}{2} = 3 \pm 22$ — Critical values by the quadratic formula. $32 = 42$ .
6
$k < 3 - 22 or k > 3 + 22$ — Greater than 0: the two outside parts.

Rule

0606 and 4037 Paper 1 has no calculator. Leave critical values as simplified surds, like $3 \pm 22$ . A decimal can lose the final mark when the question asks for exact form.

Worked example

1
Find the set of values of c for which the line $y = 3 x + c$ does not meet the curve $y = 2 x^{2} - x + 5$ .
2
$2 x^{2} - 4 x + 5 - c = 0$ — Substitute and collect. The number on its own is the whole of $5 - c$ .
3
$(- 4)^{2} - 4 (2) (5 - c) < 0$ — Does not meet: discriminant less than 0.
4
$16 - 40 + 8 c < 0$ — Expand, keeping the bracket.
5
$c < 3$ — So $8 c < 24$ .

Common mistake

Mixing up the c in the line with the c in $b^{2} - 4 a c$ . Here the number on its own is the whole of $5 - c$ .

Key idea

Two points: greater than 0. Misses: less than 0. Meets at all: greater than or equal to 0.

Other curves

The curve does not have to start with $y =$ . Substitute whichever letter makes the working shortest. The test is the same.

Worked example

1
Find the value of m for which the line $y = m x + 1$ is a tangent to the curve $y^{2} = 4 x$ .
2
$(m x + 1)^{2} = 4 x$ — Replace y in the curve with the line.
3
$m^{2} x^{2} + 2 m x + 1 - 4 x = 0$ — Expand the square and move 4x across.
4
$m^{2} x^{2} + (2 m - 4) x + 1 = 0$ — Group the x terms. Now $a = m^{2}$ , $b = 2 m - 4$ , $c = 1$ .
5
$(2 m - 4)^{2} - 4 m^{2} = 0$ — Tangent: discriminant equals 0.
6
$16 - 16 m = 0$ — The $4 m^{2}$ terms cancel.
7
$m = 1$ — So the tangent is $y = x + 1$ .

Example

Check: with $m = 1$ the equation is $x^{2} - 2 x + 1 = 0$ , so $x = 1$ and $y = 2$ . The point (1, 2) is on $y^{2} = 4 x$ , because $2^{2} = 4 \times 1$ .

Rule

“Show that the x-coordinates satisfy …” means the expanding and collecting lines are the marks. Write each line; do not jump to the given answer.

Common mistake

Reading a as 1 when the $x^{2}$ term has a letter in front. Here $a = m^{2}$ , so $- 4 a c = - 4 m^{2}$ .

Key idea

Any line, any curve: substitute, collect to 0, then read a, b and c from that new equation.

Which test? Read the question

Read the question first. The words tell you which sign to put on the discriminant.

You see	You do
“Is a tangent” or “touches”	$b^{2} - 4 a c = 0$
“Meets at two distinct points” or “intersects twice”	$b^{2} - 4 a c > 0$
“Does not meet” or “does not intersect”	$b^{2} - 4 a c < 0$
“Meets the curve”	$b^{2} - 4 a c \geq 0$
“Find the point of contact”	Solve the repeated root, then use the line for y
“Show that … and hence find k”	Show each step of the substitution, then use the discriminant

For 0606 and 4037, the circle lesson uses this same test when a line meets a circle.

Common mistake

Writing the answer in x. The question wants the values of k (or m, or c) that make the line behave this way.

Key idea

Number of meeting points = number of real roots. The discriminant counts them for you.

Practise when a line meets, touches or misses a curve now

Interactive visual aids, then real past-paper questions marked on your working like the exam. Free to start.

Practise free on Lemmara

Real past-paper questions on when a line meets, touches or misses a curve

Each with a full worked solution.

Lemmara is an independent study tool, not affiliated with or endorsed by Cambridge International Education or Pearson Edexcel. The lessons and worked examples are Lemmara’s own.