Question

a.   Find the values of the constant m for which the line  is a tangent to the curve .

b.   The function f is defined for  by , where a and b are constants. The solutions of the equation  are x = 1 and x = 9. Find

i.       the values of a and b,

 ii.       y=the coordinates of the vertex of the curve .

Solution

a.
 

We are given that line with equation  is tangent to the curve with equation;

Since line is tangent to the curve, both intersect at only a single point.

We need to find coordinates of point of intersection to find the slope of line.

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;

We need to solve this equation to find the values of .

We recognize that it is a quadratic equation.

Standard form of quadratic equation is;

Expression for discriminant of a quadratic equation is;

If   ; Quadratic equation has two real roots.

If   ; Quadratic equation has no real roots.

If   ; Quadratic equation has one real root/two equal roots.

Since line is tangent to the curve, they intersect at only a single point and, hence, solution of  above equation has only one real (or repeated/equal) root(s). Therefore;

For the equation ;

Therefore;

Now we have two options.

b.
 

We are given that function f is defined for  by  and solutions of the  equation  are x = 1 and x = 9.

Therefore solutions of the equation  are x = 1 and x = 9.

i.
 

We can also write the factors of equation from given roots;

Comparing with the equation for which solutions were given ;

Therefore;

ii.
 

We have found from (b:i) that

is a quadratic equation.

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 given curve , is a parabola opening upwards.

Vertex form of a quadratic equation is;

The given function  can be written in vertex form as follows.

We have the expression;

We use method of “completing square” to obtain the desired form.

Next we complete the square for the terms which involve .

We have the algebraic formula;

For the given case we can compare the given terms with the formula as below;

Therefore we can deduce that;

Hence we can write;

To complete the square we can add and subtract the deduced value of ;

Therefore;

Coordinates of the vertex are .Since this is a parabola opening upwards the vertex is the minimum point on the graph.
Here y-coordinate of vertex represents least value of  and x-coordinate of vertex represents  corresponding value of .

For the given case, vertex is .