Cambridge Additional Mathematics

186 Straight line graphs (Chapter 7)

To find where a straight line intersects a curve, we first rearrange the equation of the line so thatxoryis

the subject. We then substitute this expression forxoryinto the equation of the curve.

While a straight line meets another straight line at most once, a straight line may meet a curve more than

once.

Example 10 Self Tutor

Find the points where the line x¡ 3 y=4intersects the curve x^2 +y^2 =34.

Substituting x=3y+4 into x^2 +y^2 =34 gives

(3y+4)^2 +y^2 =34

) 9 y^2 +24y+16+y^2 =34

) 10 y^2 +24y¡18 = 0

) 2(5y^2 +12y¡9) = 0

) 2(5y¡3)(y+3)=0

) y=^35 or¡ 3

When y=^35 , x=3(^35 )+4=^295

When y=¡ 3 , x=3(¡3) + 4 =¡ 5

) the line intersects the curve at (^295 ,^35 ) and (¡ 5 ,¡3).

Example 11 Self Tutor

Find the points where the line 2 x+3y=5intersects the curve

1

x

¡

3

y

=2.

If 2 x+3y=5, then y=

5 ¡ 2 x

3

.

Substituting into

1

x

¡

3

y

=2 gives

1

x

¡

3

5 ¡ 2 x

3

=2

)

1

x

¡

9

5 ¡ 2 x

=2

) (5¡ 2 x)¡ 9 x=2x(5¡ 2 x) f£both sides by x(5¡ 2 x)g

) 5 ¡ 11 x=10x¡ 4 x^2

) 4 x^2 ¡ 21 x+5=0

) (4x¡1)(x¡5) = 0

) x=^14 or 5

When x=^14 , y=

5 ¡2(^14 )

3

=^32 , and when x=5, y=

5 ¡2(5)

3

=¡^53.

) the line intersects the curve at (^14 ,^32 ) and (5,¡^53 ).

C Intersection of a straight line and a curve

CURVE

C

AND A

y

O x

x+y=34 22

x-3y=4

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Additional Mathematics
Y:\HAESE\CAM4037\CamAdd_07\186CamAdd_07.cdr Friday, 20 December 2013 3:30:30 PM BRIAN