Advertisement Remove all ads

Prove that for an Astroid X 2 3 + Y 2 3 = a 2 3 the Line 𝜽=𝝅/𝟔 Divide the Arc in the First Quadrant in a Ratio 1:3. - Applied Mathematics 2

Prove that for an astroid  `   x^(2/3) +y2/3= a^(2/3)` the line 𝜽=𝝅/𝟔 Divide the arc in the first quadrant in a ratio 1:3. 


Advertisement Remove all ads


Given curve : astroid` x^(2/3)+y^(2/3) = a^(2/3)`

The line 𝜽=𝝅/𝟔 cuts the asroid in 1 st quadrant. 

C is the point on the curve which cuts the arc.
Length of astroid in first quadrant: 

Put `  x = acos^3t and y=asin^3t` 

`dx=-3 asin t.cos^2tdt    dy=3 acos t.sin^2tdt`

`s= int_0^(pi/2) sqrt ((dx/dt)^2+(dy/dt)^2)=int_0^(pi/2) sqrt((-3asin t.cos^2 t)^2+(3 acos t .sin^2 t )^2)` dt 

= `int_0^(pi/2) 3a.sin t.cost dt` 

= `3/2 a  int_0^(pi/2) sin 2t  dt` 

=`3/4 a [-cos 2t ]_0^(pi/2) `

∴ `s= 3/2 a `                        ………………….(1) 

Now the length of the curve ac : Just put `pi/6` 𝒊𝒏𝒔𝒕𝒆𝒅 𝒐𝒇 `pi/2`  because the curve is Only upto given line. 

∴ S(ac) =`int_0^(pi/6) 3a sint . cost dt =3/4a [-cos 2t]_0^(pi/6) ` 

=` 3/4 a [-1/2+1]` 

`s(ac)= 3/8 a  `                              ……………(2) 

Legnth of remaining part = `3/2a-3/8 a=9/8 a`  ……………….(3)

Divide eqn (3) and (2).
The line `pi/6` cuts the given astroid in the ratio of 1:3
Hence proved.

Concept: Exact Differential Equations
  Is there an error in this question or solution?
Advertisement Remove all ads


Advertisement Remove all ads
Advertisement Remove all ads

View all notifications

      Forgot password?
View in app×