Cambridge International AS and A Level Mathematics Pure Mathematics 1

Exercise 2F

P1^

2

You can use the discriminant, b^2 – 4ac, to find the value of k such that the

equation has one repeated root. The condition is b^2 – 4ac=    0

 y^2 − 4 y + 2 k = 0 ⇒ a = 1, b = −4 and c = 2 k

 b^2 − 4 ac = 0 ⇒ (−4)^2 − 4 × 1 × 2 k = 0

⇒ 16 − 8 k = 0

⇒ k= 2

So the line 2 y = x + 2 forms a tangent to the curve y^2 = 2x.

(ii) You have already started to solve the equations 2y = x + 2 and y^2 = 2x in

part (i). Look at equation ^4 : y^2 − 4 y + 2 k = 0

You know from part (i) that k = 2 so you can solve the quadratic to find y.

 y^2 − 4 y + 4 = 0

⇒ (y − 2)(y − 2) = 0

⇒ y = 2

Notice that this is a repeated root so the line is a tangent to the curve.

Now substitute y = 2 into the equation of the line to find the x co-ordinate.

When y = 2: 2 y = x + 2 ⇒ 4 = x + 2

 x = 2

So the tangent meets the curve at the point (2, 2).

EXERCISE 2F 1 Show that the line y = 3 x + 1 crosses the curve y = x^2 + 3 at (1, 4) and find the
co-ordinates of the other point of intersection.
2 (i) Find the co-ordinates of the points A and B where the line y = 2 x − 1 cuts
the curve y = x^2 − 4.
(ii) Find the distance AB.
3 (i) Find the co-ordinates of the points of intersection of the line y = 2 x and
the curve y = x^2 + 6 x − 5.
(ii) Show also that the line y = 2 x does not cross the curve y = x^2 + 6 x + 5.
4 The line 3y = 5 − x intersects the curve 2y^2 = x at two points. Find the distance
between the two points.
5 The equation of a curve is xy = 8 and the equation of a line is 2x + y = k, where
k is a constant. Find the values of k for which the line forms a tangent to the
curve.
6 Find the value of the constant c for which the line y = 4x + c is a tangent to the
curve y^2 = 4x.