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Question 13
Circle $$C_{1}$$ has a radius of 3 units. The line segment PQ is the only diameter of the circle which is parallel to the X axis. P and Q are points on curves given by the equations $$y = a^{x} and y = 2a^{x}$$ respectively, where a < 1. The value of a is:
Solution
Radius = 3 units, => Diameter = PQ = 6 units
y-coordinates of P and Q are same as PQ is parallel to x-axis
x-coordinates of P and Q will have a difference of 6 units.
Equating y-coordinate of P and Q
=> $$a^x = 2 a^{x + 6}$$
=> $$\frac{1}{2} = \frac{a^{x + 6}}{a^x}$$
=> $$a^{x + 6 - x} = \frac{1}{2}$$
=> $$a = \frac{1}{\sqrt[6]{2}}$$
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