Question

Find the equation of tangent at the point \( (am^2, am^3) \) on the curve \[ ay^2 = x^3. \]

Hint:

The equation of the tangent line at a point on a curve can be found using the point-slope form: \[ y - y_1 = m(x - x_1), \] where \( m \) is the slope of the tangent and \( (x_1, y_1) \) is the point of tangency.

Answer 1

The given curve is \[ ay^2 = x^3. \] Differentiating both sides with respect to \( x \), using implicit differentiation: \[ \frac{d}{dx}(ay^2) = \frac{d}{dx}(x^3), \] \[ 2ay \frac{dy}{dx} = 3x^2. \] Thus, \[ \frac{dy}{dx} = \frac{3x^2}{2ay}. \] Now, substitute \( x = am^2 \) and \( y = am^3 \) into this equation to find the slope of the tangent at the point \( (am^2, am^3) \): \[ \frac{dy}{dx} = \frac{3(am^2)^2}{2a(am^3)} = \frac{3a^2m^4}{2a^2m^3} = \frac{3m}{2}. \] The equation of the tangent line at \( (am^2, am^3) \) is given by the point-slope form: \[ y - am^3 = \frac{3m}{2}(x - am^2). \] Conclusion: The equation of the tangent line is \[ \boxed{y - am^3 = \frac{3m}{2}(x - am^2)}. \]