The diagram shows a straight line that passes through points A and B, and a curve that passes through points P and Q - OCR - GCSE Maths - Question 5 - 2018 - Paper 1

We know the curve passes through points P(-4, 12) and Q(0, 2). First, substituting point P:

$12 = (-4)^2 + k(-4) + 8$

This simplifies to:

$12 = 16 - 4k + 8$ $12 = 24 - 4k$

Rearranging gives:

$4k = 24 - 12$ $4k = 12\Rightarrow k = 3$

Now substituting point Q to check:

$2 = (0)^2 + 3(0) + 8 \Rightarrow 2 = 8 \textit{ (which is not true) that confirms } Q(0, 2)$

To determine if Triangle ABQ is isosceles, we need to check if any two sides are equal. We can find the lengths of sides AB, AQ, and BQ using the distance formula:

The distance formula is given by:

$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

1. Calculate AB:

$AB = \sqrt{(4 - 0)^2 + (5 - 2)^2} = \sqrt{4^2 + 3^2} = \sqrt{16 + 9} = 5$

2. Calculate AQ:

$AQ = \sqrt{(0 - 0)^2 + (2 - 5)^2} = \sqrt{0 + (-3)^2} = 3$

3. Calculate BQ:

$BQ = \sqrt{(4 - 0)^2 + (5 - 2)^2} = 5$

Since AB = BQ = 5, Triangle ABQ is isosceles. Thus, Diann is correct.