Cambridge Additional Mathematics

92 Quadratics (Chapter 3)

Example 25 Self Tutor

y=2x+k is a tangent to y=2x^2 ¡ 3 x+4. Findk.

y=2x+k meets y=2x^2 ¡ 3 x+4 where

2 x^2 ¡ 3 x+4=2x+k

) 2 x^2 ¡ 5 x+(4¡k)=0

Since the graphs touch, this quadratic has ¢=0

) (¡5)^2 ¡4(2)(4¡k)=0

) 25 ¡8(4¡k)=0

) 25 ¡32 + 8k=0

) 8 k=7

) k=^78

2 For which value ofcis the line y=3x+c a tangent to the parabola with equation

y=x^2 ¡ 5 x+7?

3 Find the values ofmfor which the lines y=mx¡ 2 are tangents to the curve with equation

y=x^2 ¡ 4 x+2.

4 Find the gradients of the lines withy-intercept 1 that are tangents to the curve f(x)=3x^2 +5x+4.

5aFor what values ofcdo the lines y=x+c never meet the parabola with equation

y=2x^2 ¡ 3 x¡ 7?

b Choose one of the values ofcfound in partaabove. Illustrate with a sketch that these graphs

never meet.

6 Consider the curve y=x^2 +4x¡ 1 and the line

y=2x+c. Find the value(s) ofcfor which the

line:

a meets the curve twice

b is a tangent to the curve

c does not meet the curve.

7 Consider the curve f(x)=¡x^2 +3x¡ 6 and

the line g(x)=mx¡ 2. Find the values ofm

for which the line:

a meets the curve twice

b is a tangent to the curve

c does not meet the curve.

y

O x

y=2x+c

y=x +4x-1 2

c

y

O x

y=mx-2

y=-x +3x-6 2

-2

A line which is a tangent to a

quadratic willtouchthe curve.

DEMO

DEMO

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Y:\HAESE\CAM4037\CamAdd_03\092CamAdd_03.cdr Monday, 20 January 2014 3:33:28 PM BRIAN