i need help for this question

the curve c has equation y = x squared + 3px + 8p where p is a real constant. the straight line L has equation y = 1 - 2x given that L does not intersect C find the set of possible values of p giving your answer in exact form

Reply 1

M0407

You need to use the discriminant.

Make both equations equal to each other and rearrange:
x^2+3px+8p=1-2x
Becomes: x^2 +3px-2x+8p-1=0

Then, since the lines don’t intersect, discriminant will be less than 0
So b^2 - 4ac < 0
Which if you sub in the equation it is: (3p-2)^2 - 4(8p-1) < 0
= 9p^2 - 12p + 4 - 32p + 4 < 0
= 9p^2 - 45p < 0
= 3p^2 - 15p < 0

Then you factorise and solve this quadratic inequality by drawing a sketch of the quadratic and then seeing where the values are below 0 which is the section under the x axis of the graph. Type up solving quadratic inequalities on YouTube and it should help you with the last step.

Hope it helps!

Reply 2

mqb2766