A-LEVEL•Mathematics•Quadratics•Question 6

Solution

🟡 Exam-style · [5 marks] The line $y = 2x + c$ is a tangent to the curve $y = x^2 - 4x + 11$. Find the value of the constant $c$ and the coordinates of the point of contact.

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Solution

Equate the line and the curve:

\begin{aligned} x^2 - 4x + 11 &= 2x + c \\ x^2 - 6x + (11 - c) &= 0 \end{aligned}

For a tangent the discriminant is zero:

\begin{aligned} 36 - 4(11 - c) &= 0 \\ 36 - 44 + 4c &= 0 \\ c &= 2 \end{aligned}

Then $x^2 - 6x + 9 = (x-3)^2 = 0$ , so $x = 3$ and $y = 2(3) + 2 = 8$ .

The point of contact is $(3,\ 8)$ . Check on the curve: $9 - 12 + 11 = 8$ ✓

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