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Show that the Length of Curve 9 a Y 2 = X ( X − 3 a ) 2 is 4 √ 3 a - Applied Mathematics 2

ConceptRectification of Plane Curves

Question

Show that the length of curve 9ay^2=x(x-3a)^2  "is"  4sqrt3a

Solution

Curve : 9ay^2=x(x-3a)^2 ................(1)
The given curve is strophoid.

Differentiate eqn (1) w.r.t x,

18ay(dy)/(dx)=2x(x-3a)+(x-3a)^2

18ay(dy)/(dx)=2x(x-3a)+(x-3a)^2

therefore(dy)/(dx)=((x-3a)(x-a))/(6ay)

Squaring both the sides,

((dy)/(dx))^2=((x-3a)^2(x-a)^2)/(36a^2y^2)

therefore((dy)/(dx))^2=((x-3a)^2(x-a)^2)/(4ax(x-3a)^2) ............from(1)

therefore((dy)/(dx))^2=(x-a)^2/(4ax)

The perimeter of given curve is ,

S=int_0^(3a)sqrt(1+((dy)/(dx))^2dx)=int_0^(3a)sqrt(1+(x-a)^2/(4ax))dx=int_0^(3a)sqrt((x+a)^2/(4ax))dx

therefore S=int_0^(3a) (x+a)/(2sqrtxsqrta)dx

therefore S=1/(2sqrta)int_0^(3a) (x+a)/sqrtxdx

=1/(2sqrta)[(2xsqrtx)/3+2sqrtx]_0^(3a)

=1/(2sqrta)[(2asqrt(3a))/1+2sqrt3a]

thereforeS=2sqrt3  ......................( Half curve length)

∴ The total length of given curve = 2 S = 4sqrt3 units.

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Solution Show that the Length of Curve 9 a Y 2 = X ( X − 3 a ) 2 is 4 √ 3 a Concept: Rectification of Plane Curves.
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