AS 91578 · 2025 paper · logs, exponentials, tangents, radius of curvature

Question 2(c) Merit

Practises Tangents, normals and curve features

A reciprocal minus a log, a product that has to be differentiated to decide increasing or decreasing, two tangents of the same gradient to a rational curve, a parametric tangent in sec and tan, and a given curvature formula to apply.

The graph shows y = f(x) = x²/(x + 4). Two tangents to this curve have the equation y = −3x + c.

Find the coordinates of the two points of contact of these tangents with the curve y = f(x).

The 2025 paper, page 6 ↗ Marking schedule, page 3 ↗

Work it out

Try it on paper first. Stuck? Take one hint at a time.

1 If y = −3x + c touches the curve, what must be true where it touches?
There the curve has the same gradient as the line: dy/dx = −3. You never need c.
2 Differentiate with the quotient rule.
dy/dx = (2x(x + 4) − x² · 1)/(x + 4)² = (x² + 8x)/(x + 4)²
3 Set it equal to −3 and solve.
x² + 8x = −3(x² + 8x + 16)
4x² + 32x + 48 = 0
x² + 8x + 12 = (x + 2)(x + 6) = 0, so x = −2 or x = −6.
4 The question wants coordinates. Find y for each.
x = −2: y = 4/2 = 2
x = −6: y = 36/(−2) = −18
One point on each branch of the graph, as the sketch suggests.

Answer (−2, 2) and (−6, −18).

What the marker wanted

Achieved The correct derivative.
Merit Both points of contact, x and y for each, worked out from the derivative.

In our words, from NZQA’s 2025 marking schedule.

Common mistake

Not seeing that "tangent with gradient −3" means "set the derivative to −3". NZQA’s 2025 report names this part, with One (d), as one where many students could not join their differentiation to the coordinate geometry.

How did it go?

Got it moves it further out. Review brings it back tomorrow.