8. WMA11/01 Edexcel IAL P1 Oct 2021 Q8 (Quadratics)
Figure 4 shows a sketch of the curve C with equation
The point M is the maximum turning point on C.
(a)
(i) Write
where a, b and c are constants to be found.
(ii) Hence, or otherwise, state the coordinates of M.
(5)
The line l1 passes through O and M, as shown in Figure 4.
A line l2 touches C and is parallel to l1
(b) Find an equation for l2
(5)
SOLUTION
a-
i- Using completing square method to express the quadratic equation in the given form.
ii-
The quadratic equation can be expressed in the form of completing square form as
In this question, the vertex of a quadratic equation is given as (2, 16).
b-
Since l1 and l2 are parallel, there gradients are equal to each other. So, first we will find the gradient of line l1.
The gradient of l1 as it passes through O (0,0) & M (2,16).
So, the gradient of line l2 is
Now, the equation of line l2 is
As it is given that line l2 touches C, it means the line l2 cuts curve at a single point, or in ither words, line l2 is tangential to the curve. Hence, solving both equations simultaneously.
Since the line touches curve or is tangential, so the above simultaneously solved equation must have one solution, which means the discriminant is equal to 0.
Putting this value of c in the equation we made earlier of line l2.
