Question

All points on the curve \({y^2} = 4a\left( {x + a\sin \frac{x}{a}} \right)\) at which the tangents are parallel to the axis of x, lie on a

1. Circle
2. parabola
3. Line
4. None of these

Answer 1

Correct Answer - Option 2 : parabola

\({y^2} = 4a\left( {x + a\sin \frac{x}{a}} \right)\)     ---(1)

Differentiating the above, we get:

\({2y}\frac{dy}{dx} = 4a\left( {1 + \cos \frac{x}{a}} \right)\)   ---(2)

If the tangent is parallel to the x-axis, dy/dx = 0

Putting this value in Equation (2), we get:

cos (x/a) = -1

∴ sin (x/a) = 0

On putting this value in Eq. (1), we get:

y² = 4a (x + 0)

y² = 4ax

This is the equation of a parabola.