If the curve y = a x and y = b x intersect at angle α , then tan α is equal to

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TestHub · Accepted Answer

Clearly, the point of intersection of curves is $(0, 1)$ .

Now, slope of tangent of first curve

$m_{1} = \frac{d y}{d x} = a^{x} \log a$

$\Rightarrow {(\frac{d y}{d x})}_{(0, 1)} = m_{1} = \log a$

Slope of tangent of second curve,

$m_{2} = \frac{d y}{d x} = b^{x} \log b$

$\Rightarrow m_{2} = {(\frac{d y}{d x})}_{(0, 1)} = \log b$

$∴ \tan α = |\frac{m_{1} - m_{2}}{1 + m_{1} m_{2}}| = \frac{\log a - \log b}{1 + \log a \log b}$

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