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Maths-

General

Easy

Question

If $S T$ and $S N$ are the lengths of the subtangent and the subnormal at the point $theta equals fraction numerator pi over denominator 2 end fraction$ on the curve $x equals a open parentheses theta plus sin invisible function application theta close parentheses comma blank y equals a open parentheses 1 minus cos invisible function application theta close parentheses comma blank a not equal to 1$ , then

$S T = S N$
$S T = 2 S N$
$S T^{2} = a S N^{3}$
$S T^{3} = a S N$

The correct answer is: $S T equals S N$

Given,
$x equals a left parenthesis theta plus sin invisible function application theta right parenthesis$ and $y equals a left parenthesis 1 minus cos invisible function application theta right parenthesis$

$rightwards double arrow blank fraction numerator d x over denominator d theta end fraction equals a left parenthesis 1 plus cos invisible function application theta right parenthesis$ and $fraction numerator d y over denominator d theta end fraction equals a sin invisible function application theta$

$therefore blank fraction numerator d y over denominator d x end fraction equals fraction numerator a sin invisible function application theta over denominator a left parenthesis 1 plus cos invisible function application theta right parenthesis end fraction equals tan invisible function application fraction numerator theta over denominator 2 end fraction$

Now, length of subtangent $equals open vertical bar fraction numerator y over denominator d y divided by d x end fraction close vertical bar$

$therefore blank S T equals fraction numerator a left parenthesis 1 minus cos invisible function application theta right parenthesis over denominator tan invisible function application left parenthesis theta divided by 2 right parenthesis end fraction equals a sin invisible function application theta$

$rightwards double arrow$ Length of suntangent at $theta equals fraction numerator pi over denominator 2 end fraction$ ,

$S T equals a sin invisible function application fraction numerator pi over denominator 2 end fraction equals a$

And length of subnormal $equals open vertical bar y fraction numerator d y over denominator d x end fraction close vertical bar$

$rightwards double arrow blank S N equals a open parentheses 1 minus cos invisible function application theta close parentheses. tan invisible function application fraction numerator theta over denominator 2 end fraction$

$equals a.2 sin to the power of 2 end exponent invisible function application fraction numerator theta over denominator 2 end fraction tan invisible function application fraction numerator theta over denominator 2 end fraction$

$rightwards double arrow$ Length of subnormal at $theta equals fraction numerator pi over denominator 2 end fraction$ ,

$S N plus a.2.1.2 equals a$

Hence, $S N equals S T$

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