# Past Papers’ Solutions | Edexcel | AS & A level | Mathematics | Core Mathematics 1 (C1-6663/01) | Year 2009 | June | Q#11

Question

The curve C has equation

The point P has coordinates (2, 7).

a)   Show that P lies on C.

b)  Find the equation of the tangent to C at P, giving your answer in the form y=mx+c, where m and c  are constants.

The point Q also lies on C.

Given that the tangent to C at Q is perpendicular to the tangent to C at P,

c)   show that the x-coordinate of Q is

Solution

a)

If a point lies on the curve (or the line), the coordinates of that point satisfy the equation of the curve  (or the line).

We are given equation of the curve C as;

We are also given coordinates of the point P(2,7).

We substitute x-coordinate of point P in the equation of the curve C;

Since coordinates of the point P satisfy the equation of the curve C, therefore the point lies on the curve.

b)

We are required to find the equation of tangent to the curve C at point P(2,7).

To find the equation of the line either we need coordinates of the two points on the line (Two-Point  form of Equation of Line) or coordinates of one point on the line and slope of the line (Point-Slope  form of Equation of Line).

We already have coordinates of a point on the tangent as P(2,7).

We need to find slope of tangent at in order to write its equation.

The slope of a curve  at a particular point is equal to the slope of the tangent to the curve at the same point;

Therefore, if we can find slope of the curve C at point P(2,7) then we can find slope of the tangent to  the curve at this point.

We need to find the gradient of the curve C at point P(2,7).

Gradient (slope) of the curve at the particular point is the derivative of equation of the curve at that  particular point.

Gradient (slope)  of the curve  at a particular point  can be found by substituting x- coordinates of that point in the expression for gradient of the curve;

We are given equation of the curve as;

Therefore;

Rule for differentiation is of  is:

Rule for differentiation is of  is:

Rule for differentiation is of  is:

For gradient of the curve at point P(2,7), substitute  in derivative of the equation of the curve.

Therefore;

With coordinates of a point on the tangent P(2,7) and its slope  in hand, we can write equation of the tangent.

Point-Slope form of the equation of the line is;

c)

We are given that tangent to the curve C at point Q is perpendicular to the tangent to the curve C at  point P.

If a line  is normal to the curve , then product of their slopes  and  at that point (where line is normal to the curve) is;

Therefore;

Slope of Tangent at point P X Slope of tangent at point
Q=-1

We have found in (b) that;

Therefore;

Gradient (slope) of the curve is the derivative of equation of the curve. Hence gradient of curve  with respect to  is:

We have from (b) that;

If we have coordinates of a point on the curve, we can find gradient of the curve at that point by  substituting coordinates of this point in expression of the derivative of equation of the curve.

Conversely, if we have gradient of the curve at particular, we can equate expression of the  derivative of equation of the curve with the given gradient of curve at that point to find the  coordinates of that point.

Therefore;

For a quadratic equation , the expression for solution is;

Therefore;

Since ;

Now we have two options.