Maths-
General
Easy

Question

If the relation between subnormal 'SN' and subtangent 'ST' at any point on the curve by space Y squared equals left parenthesis x plus a right parenthesis cubed is p left parenthesis S N right parenthesis equals q space left parenthesis S T right parenthesis squared then p over q equals

  1. fraction numerator 8 over denominator 27 a end fraction
  2. 8 over 27
  3. fraction numerator 8 over denominator 27 b end fraction
  4. fraction numerator 8 b over denominator 27 end fraction

hintHint:

We are given a curve. We are given the relation between the subtangent and the subnormal. We have to find the ratio of the variables p and q. We will first find the values of subtangent and subnormal using the formulae. Then we will find the ratio.

The correct answer is: fraction numerator 8 b over denominator 27 end fraction


    The given curve is y squared equals open parentheses x plus a close parentheses cubed.
    The relation between subtangent "ST" and subnormal is "SN" is p(SN) = q(ST)2.
    We have to find the ratio of p over q.
    Subtangent: It is the projection of tangent on x-axis from the x intercept to point of tangency.
    Subnormal: It is the projection of normal on x-axis from the x intercept to point of tangency.
    The formula of subtangent is ST= fraction numerator y over denominator fraction numerator d y over denominator d x end fraction end fraction
    The formula of subnormal is SN = y fraction numerator d y over denominator d x end fraction
    we will differentiate the curve w.r.t x
    y squared space equals left parenthesis x space plus space a right parenthesis cubed space
2 y fraction numerator d y over denominator d x end fraction equals space 3 left parenthesis x space plus space a right parenthesis squared left parenthesis 1 right parenthesis space space space space space space space
space space space fraction numerator d y over denominator d x end fraction space equals fraction numerator 3 left parenthesis x space plus space a right parenthesis squared over denominator 2 y end fraction

N o w comma space S T space equals space fraction numerator y over denominator begin display style fraction numerator 3 left parenthesis x space plus space a right parenthesis squared over denominator 2 y end fraction end style end fraction
space space space space space space space space space space space space space space space equals fraction numerator 2 y squared over denominator 3 left parenthesis x space plus space a right parenthesis squared end fraction
space space space space space space space space space space space space space space equals fraction numerator 2 left parenthesis x space plus space a right parenthesis cubed over denominator 3 left parenthesis x space plus space a right parenthesis squared end fraction
space space space space space space space space space space space space space space space equals space 2 over 3 left parenthesis x space plus space a right parenthesis
left parenthesis S T right parenthesis to the power of 2 space end exponent equals 4 over 9 left parenthesis x space plus space a right parenthesis squared
    S N space equals space y space cross times fraction numerator 3 left parenthesis x space plus space a right parenthesis squared over denominator 2 y end fraction
space space space space space space equals 3 over 2 left parenthesis x space plus space a right parenthesis squared
    Now,  p(SN) = q(ST)2
    p space cross times space 3 over 2 left parenthesis x space plus space a right parenthesis squared equals q cross times fraction numerator space 4 over denominator 9 end fraction left parenthesis x space plus space a right parenthesis squared space space space space space space space space space space space space space space space space
space space space space space space space space space space space space space space space space space space space space space space p over q space equals space 4 over 9 space cross times space 2 over 3
space space space space space space space space space space space space space space space space space space space space space p over q space equals 8 over 27

    For such questions, we should know the formulas of subnormal and subtangent.

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