MATHEMATICS

The intersection of two curves
PS
PS
PS
9 (i) Find the value of k for which the line 2y = x + k forms a tangent to the
curve y 2 = 2x.
(ii) Hence find the coordinates of the point where the line 2y = x + k
meets the curve for the value of k found in part (i).
10 The equation of a circle is (x + 2) 2 + y 2 = 8 and the equation of a line
is x + y = k, where k is a constant.
Find the values of k for which the line forms a tangent to the curve.
11 The equations of two circles are given below
( x + 1) 2 + ( y − 2) 2 = 10
and ( x − 1) 2 + ( y − 3) 2 = 5
The two circles intersect at the points A and B.
Find the area of the triangle AOB where O is the origin.
KEY POINTS
1 For a line segment A(x 1
, y 1
) and B(x 2
, y 2
) (Figure 5.37) then:
n The gradient of AB is y − y
x − x
y
2 1
2 1
B (x 2
, y 2
)
A
(x 1
, y 1
)
O
x
n
Figure 5.37
The midpoint is
x + x y + y
,
2 2
1 2 1 2
2
n The distance AB is ( x − x ) + ( y − y )
2 1
2
2 1
Using Pythagoras’
theorem.
2 Two lines are parallel ➝➝ gradients are equal.
3 Two lines are perpendicular ➝➝ the product of their gradients is −1.
4 The equation of a straight line may take any of the following forms:
n line parallel to the y axis: x = a
n line parallel to the x axis: y = b
n line through the origin with gradient m: y = mx
n line through (0, c) with gradient m: y = mx + c
n line through (x 1
, y 1
) with gradient m: y − y 1
= m(x − x 1
)
5 The equation of a circle is
n centre (0, 0), radius r: x 2 + y 2 = r 2
n centre (a, b), radius r: (x − a) 2 + (y − b) 2 = r 2 .
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