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For a curve $y = f (x)$ , if $a, b, l, m$ represents the lengths of tangent normal, subtangent, sub normal respectively then $a^{2} : b^{2} =$

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$l : m$

$m : l$

$l^{2} : m^{2}$

$m^{2} : l^{2}$

answer is A.

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Detailed Solution

$\frac{a^{2}}{b^{2}} = \frac{\frac{y_{1}^{2}}{m^{2}} (1 + m^{2})}{y_{1}^{2} (1 + m^{2})} = \frac{1}{m^{2}}$
$\frac{l}{m} = \frac{\frac{y_{1}}{m}}{y_{1} m} = \frac{1}{m^{2}}$
$a^{2} : b^{2} = l : m$

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