All points on the curve `y^(2)=4a(x+a" sin"(x)/(a))` at which the tangents are parallel to the axis of x lie on a

Options

A. circle
B. parabola
C. straight line
D. None of these

Answer

B

Explanation

` y ^ 2 = 4a [ x + a sin (( x ) /( a ))]" " `…(i)
` therefore 2y ( dy )/ ( dx ) = 4a [ 1 + cos ( ( x )/(a )) ]" " `… (ii)
If tangent is parallel to x - axis , then
` (dy ) /( dx ) = 0 `
So, from Eq. (i), we get
` cos (( x )/( a )) = - 1 `
` therefore sin (( x )/( a ))= 0 `
On putting this value in Eq. (i) , we get
` y^ 2 = 4a ( x + 0 ) rArr y ^ 2 = 4a x `
So, all the points on the curve
` y^ 2 = 4a ( x + a sin "" ( x ) /( a )) `,
where the tangents is parallel to the x-axis are lies on parabola.