3 Unit A2 1: Pure Mathematics (A-level only)
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Question 209
Figure 1 below shows the curve with equation
x2−2xy+4y2−12=0x^2 - 2xy + 4y^2 - 12 = 0x2−2xy+4y2−12=0
a.[3]
Show that
dydx=x−yx−4y\displaystyle \frac{dy}{dx} = \frac{x - y}{x - 4y}dxdy=x−4yx−y
At the point P P\,P on the curve the tangent is parallel to the yyy-axis, and at the point Q Q\,Q on the curve the tangent is parallel to the xxx-axis. The positions of P P\,P and Q Q\,Q are shown in Figure 1.
b.[5]
Show that the exact distance PQ PQ\,PQ is k5k\sqrt{5}k5, where k k\,k is a rational number to be determined.