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Question

The length of tangent, subtangent, normal and subnormal for the curve $y equals x to the power of 2 end exponent plus x minus 1$ at (1, 1) are $A comma blank B comma blank C$ and $D$ respectively, then their increasing order is

$B, D, A, C$
$B, A, C, D$
$A, B, C, D$
$B, A, D, C$

The correct answer is: $B comma blank A comma blank D comma blank C$

Given, $y equals x to the power of 2 end exponent plus x minus 1$
$therefore blank fraction numerator d y over denominator d x end fraction equals 2 x plus 1$

At (1, 1), $fraction numerator d y over denominator d x end fraction equals 3 equals m$

$therefore$ Length of tangent,

$A equals open vertical bar fraction numerator y subscript 1 end subscript square root of 1 plus m to the power of 2 end exponent end root over denominator m end fraction close vertical bar equals open vertical bar fraction numerator 1 square root of 1 plus 9 end root over denominator 3 end fraction close vertical bar equals fraction numerator square root of 10 over denominator 3 end fraction$

Length of subtangent, $B equals open vertical bar fraction numerator y subscript 1 end subscript over denominator m end fraction close vertical bar equals fraction numerator 1 over denominator 3 end fraction$

Length of normal,

$C equals open vertical bar y subscript 1 end subscript square root of 1 plus m to the power of 2 end exponent end root close vertical bar equals open vertical bar 1 square root of 1 plus 9 end root close vertical bar equals square root of 10$

And length of subnormal, $D equals open vertical bar y subscript 1 end subscript m close vertical bar equals 3$

Now, increasing order is $B comma blank A comma blank D comma blank C$ .

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