The tangent or normal meets something

In 3 of 20 recent exams
Paper 1 or 2
3 to 8 marks

How it is set in the exam

Paper 1 (no calculator) Section B and Paper 2 (calculator), 3–8 marks per part. Typical: the later parts of a chain that starts with $f'(x)$ and the tangent or normal.

Spot it in the exam

Exam phrases to spot (8)

“ $L$ meets the x-axis”
“the normal meets the curve again at $Q$ ”
“cuts the curve again”
“intersects the graph at a second point”
“find the area of triangle $OAB$ ”
“the tangents at $P$ and $Q$ meet at $R$ ”
“show that $L$ passes through the point …”
“find the distance $PQ$ ”

In disguise (2)

"the coordinate axes and the tangent enclose a region": a triangle, found from the intercepts, with no integration
a general point $P(p, f(p))$ , with the answer in terms of $p$ : "show that the area does not depend on $p$ "

Looks like it, but isn't (3)

finding the tangent or normal itself: Tangent or normal at a given point
a tangent through a point off the curve, with the point of contact unknown: Find the point of contact
the area between the curve and the tangent: integration (Area between two curves)

Worked example

A tangent to a cubic crosses it again

Paper 2 (calculator)

Question

Let $f(x) = x^3 - 3x^2 + 2$ .
(a) Find the equation of the tangent to the graph of $f$ at the point where $x = -1$ . [4]
(b) The tangent meets the graph of $f$ again at the point $Q$ . Find the coordinates of $Q$ . [3]

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See it move: Where the tangent meets the curve again

At $x = 0.50$ : the tangent is $y = -2.25x + 2.5$ , and it meets the curve again at $Q(2, -2)$ .
A tangent touches where it starts, so $x = a$ is a repeated root of $f(x) =$ the tangent. The other root is the new point.
The three roots of the cubic add to $3$ , so the second meeting is at $x = 3 - 2a$ : it moves the other way as you drag.
At $x = -1$ : $y = 9x + 7$ , and $x^{3} - 3x^{2} - 9x - 5 = 0$ has $-1$ twice and $5$ , so $Q(5, 52)$ .

Which method?

The second condition decides the method. Most frequent first:

meets the curve again
Do
set line $=$ curve and rearrange to $= 0$ . The known $x$ is a root (a double root for a tangent): factor it out. : a quadratic (clear any denominator first), or a cubic that factorises on sight or that the question factorises for you, such as $(x - 1)^3$ . Any other cubic is and the GDC (solve a polynomial equation)
Watch out
the answer is the other root, then its $y$
meets another line or an axis
Do
solve the two line equations together; for an axis, put $y = 0$ or $x = 0$ in the line
Watch out
not in the curve
passes through a named point
Do
substitute the point into the line
Watch out
in a "show that", show both sides are equal
makes a triangle, or a distance is asked
Do
find the vertices first (intercepts, meeting points), then $\frac{1}{2} \times \text{base} \times \text{height}$ or the distance formula
Watch out
lengths are positive: an intercept at $x = -1$ gives a length of $1$

The line…	Do	Watch out
meets the curve again	set line $=$ curve and rearrange to $= 0$ . The known $x$ is a root (a double root for a tangent): factor it out. : a quadratic (clear any denominator first), or a cubic that factorises on sight or that the question factorises for you, such as $(x - 1)^3$ . Any other cubic is and the GDC (solve a polynomial equation)	the answer is the other root, then its $y$
meets another line or an axis	solve the two line equations together; for an axis, put $y = 0$ or $x = 0$ in the line	not in the curve
passes through a named point	substitute the point into the line	in a "show that", show both sides are equal
makes a triangle, or a distance is asked	find the vertices first (intercepts, meeting points), then $\frac{1}{2} \times \text{base} \times \text{height}$ or the distance formula	lengths are positive: an intercept at $x = -1$ gives a length of $1$

Steps

Get the line
(Tangent or normal at a given point). Keep it exact on and unrounded on . At a general point, keep $p$ as a letter.
Write the second condition as an equation
(table above).
Solve.
If the line meets the curve again, the known point is always a solution: reject it and keep the new one.
Answer what was asked:
both coordinates of a point, a length, or an area.

Common mistakes

Putting $y = 0$ into the curve
instead of into the line. The intercepts belong to the line.
Giving the known point
as the point where the line "meets the curve again". The known $x$ is always a root: the answer is the other one.
Negative lengths
an intercept at $x = -1$ gives a base of length $1$ , not $-1$ .

More mistakes (3)

Taking the new point's $y$ from $f'$
instead of from the line or the curve.
Trying to solve a cubic by hand when it doesn't factorise on sight.
On the "meets again" equation is a quadratic (for a parabola, or for $\frac{ax + b}{cx + d}$ once the denominator is cleared), or a cubic that factorises on sight or is factorised for you. Any other cubic means and the GDC.
Rounding the line on
before intersecting: use the stored values.

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