Question

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:

Accepted Answer

A. Since PQ is parallel to X-axis, y-coordinates of P and Q are equal: $a^{x_1} = 2a^{x_2} \implies a^{x_1 - x_2} = 2$. Length of diameter $PQ = 2 	imes 3 = 6 \implies x_1 - x_2 = -6$ (since $a < 1$, curve decreases, $x_1 < x_2$). $a^{-6} = 2 \implies a = 2^{-1/6} = \frac{1}{2^{1/6}}$.

Circle Diameter and Exponential Curves

Explanation

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