# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2011 | Oct-Nov | (P2-9709/23) | Q#8

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Question

The equation of a curve is 2x2 3x 3y + y2 = 6.

i.       Show that

ii.       Find the coordinates of the two points on the curve at which the gradient is 1.

Solution

i.

We are given equation of the curve as;

We are required to show that;

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

Therefore;

To find from an implicit equation, differentiate each term with respect to , using the chain rule to  differentiate any function  as .

Therefore;

Rule for differentiation of  is:

We differentiate each term of the equation, one by one, with respect to x applying following rules.

Rule for differentiation of  is:

Rule for differentiation of  is:

Now we can combines derivatives of all terms of the equation as;

ii.

We are required to find the coordinates of the two points on the curve at which the gradient is 1.

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

We have found in (i) that for the given curve.

We are given that gradient of the curve at desired points is equal to -1.

Hence;

We can substitute in given equation of the curve;

Now we have two options.

We know that ; therefore;

 For For

Hence, gradient of the given curve is -1 at two points of following coordinates.