Co-ordinate geometryP1^
2
You can find where the line and curve intersect by solving the simultaneous
equations:y − 3 x= (^2) ^1
and y = 2 x^2 ^2
Make y the subject of ^1 : y = 3 x+ (^2) ^3
Substitute ^3 into ^2 : y = 2 x^2
⇒ 3 x+ 2 = 2 x^2
⇒ 2 x^2 − 3 x− 2 = 0
⇒ ( 2 x + 1)(x– 2) = 0
⇒ x = 2 or x= –^12
Substitute into the linear equation, y = 3 x+ 2, to find the corresponding y
co-ordinates.
x = 2 ⇒ y = 8
x = –^12 ⇒ y = (^12)
So the co-ordinates of the points of intersection are (2, 8) and (–^12 ,^12 )
EXAMPLE 2.15 (i) Find the value of k for which the line 2y = x + k forms a tangent to the curve
y^2 = 2 x.
(ii) Hence, for this value of k, find the co-ordinates of the point where the line 2y
= x + k meets the curve.
SOLUTION
(i) You can find where the line forms a tangent to the curve by solving the
simultaneous equations:
2 y = x + k (^) ^1
and y^2 = 2 x (^) ^2
When you eliminate either x or y between the equations you will be left with
a quadratic equation. A tangent meets the curve at just one point and so you
need to find the value of kwhich gives you just one repeated root for the
quadratic equation.
Make x the subject of ^1 : x = 2 y− k (^) ^3
Substitute ^3 into ^2 : y^2 = 2 x
⇒ y^2 = 2(2y − k)^
⇒ y^2 = 4 y − 2 k
⇒ y^2 − 4 y + 2 k = (^0) ^4
These are the x
co-ordinates of the
points of intersection.