Question

The straight line with equation y = 3x – 7 does not cross or touch the curve with equation

y = 2px² – 6px + 4p,

where p is a constant.

a.   Show that 4p2 – 20p + 9 < 0

b.   Hence find the set of possible values of p.

Solution

a.
 

We are given that straight line and the curve do not cross or touch, hence, no intersection point. 

If two lines (or a line and a curve) intersect each other at a point then that point lies on both lines i.e.  coordinates of that point have same values on both lines (or on the line and the curve).  Therefore, we can equate  coordinates of both lines i.e. equate equations of both the lines (or the  line and the curve).

Equation of the line is;

Equation of the curve is;

Equating both equations;

Since these line and curve do not intersect there should be no solution of this equation ie equation  has no roots.

For a quadratic equation , the expression for solution is;

Where  is called discriminant.

If , the equation will have two distinct roots.

If , the equation will have two identical/repeated roots.

If , the equation will have no roots.

Since given is a quadratic equation with no real solutions of x (roots), its discriminant must be;

We have algebraic formula;

b.
 

We are required to solve the inequality;

We solve the following equation to find critical values of ;

Now we have two options;

Hence the critical points on the curve for the given condition are  & .

Standard form of quadratic equation is;

The graph of quadratic equation is a parabola. If  (‘a’ is positive) then parabola opens upwards  and its vertex is the minimum point on the graph. If  (‘a’ is negative) then parabola  opens downwards and its vertex is the maximum point on the graph.

We recognize that  it is an upwards opening parabola.

Therefore conditions for  are;