Analysis and Approaches for IBDP Maths Ebook 2 | Page 141

44
Paper 2 Section B – Gradients of Tangents and Normals
Example
A curve C is given by the implicit equation
2 2
3x 4y 2xy
48
� � � .
(a) Show that d y 3 x � y
� .
dx x � 4y
(b) The straight line L :5x � 2y
� 24 � 0 intersects C at P .
[4]
(i) Find the coordinates of P .
(c)
(ii) Hence, find the gradient of the tangent at P .
[6]
Find the coordinates of the two points on C where the gradient of the normal at
these two points is � 4.
[5]
Solution
(a)
2 2
3x � 4y � 2xy
� 48
d 2 d 2 d d
(3 x ) � (4 y ) � (2 xy) � (48)
dx dx dx dx
M1
dy
�dy�
6x �8 y � (2)( y) � (2 x) � ��
0
dx
�dx�
A2
dy
dy
6x 2y 2x 8y
dx
dx
M1
dy
3 x � y � ( x � 4 y) d x
dy 3x � y
�
dx x � 4y
AG
[4]
10
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