How to find the point of intersection for all values of m,c and a?

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Can someone please explain to me how to do question 22? Thanks!

1 Answer
Jun 15, 2018

The answer is (D). Set the two curves equal and solve for x to find points of intersection, then deduce the parameter conditions for existence of said points.

Explanation:

To find the points of intersection of the straight line y=mx+c and the parabola y=ax^2, set their y values equal and solve for x:

ax^2=mx+c
ax^2-mx-c=0
x=(m+-sqrt(m^2+4ac))/(2a)

We see from this that the curves do not intersect when the discriminant m^2+4ac<0. In this case the roots of the quadratic have an imaginary component.

Rearrange this condition:
m^2+4ac<0
4ac<-m^2

If a>0, then
c<-m^2/(4a)
If a<0, then
c> -m^2/(4a)

So the answer is (D). Your written working on the photo shows that you're very nearly there already; you just need to note that when a is negative, dividing the inequality through by it changes the inequality direction.
NB There are two parameter regions that work, but only one is listed as an option in the question - they aren't looking for the complete solution of all possible regions, rather for you to say which of their options is part of the complete solution.