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If at any point $\left(x_1, y_1\right)$ on the curve $y=f(x)$ the lengths of the subtangent and subnormal are equal, then the length of the tangent drawn to that curve at that point is
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Verified Answer
The correct answer is:
$\sqrt{2}\left|y_1\right|$
We have,
Length of subtangent $=$ Length of subnormal
$\therefore \quad y_1 \frac{d x}{d y}=y_1 \frac{d y}{d x}$
$\Rightarrow \quad \frac{d y}{d x}= \pm 1$
Length of tangent $=\left|y_1 \sqrt{1+\left(\frac{d x}{d y}\right)^2}\right|$
Length of tangent $=\left|y_1 \sqrt{1+1}\right|=\sqrt{2}\left|y_1\right|$
Length of subtangent $=$ Length of subnormal
$\therefore \quad y_1 \frac{d x}{d y}=y_1 \frac{d y}{d x}$
$\Rightarrow \quad \frac{d y}{d x}= \pm 1$
Length of tangent $=\left|y_1 \sqrt{1+\left(\frac{d x}{d y}\right)^2}\right|$
Length of tangent $=\left|y_1 \sqrt{1+1}\right|=\sqrt{2}\left|y_1\right|$
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