The parabola is .

Let  be the point on the curve.

Point  passes through the curve then .

Consider .

Apply derivative on each side with respect to .

Substitute  in the above equation.

.

Slope of the tangent line is .

Point - slope form of line equation is .

Substitute  and  in the point - slope form.

Substitute  in the above equation.

The tangent line equation is .

(a)

If the tangent line passes through the point .

Substitute  in the tangent line equation.

Substitute  in the tangent line equation.

Restrictions for :

The tangent line can never be a imaginary line.

The tangent line is reasonable for .

The tangent line passes through the point .

Therefore the tangent line is horizontal line equation .

For the parabola equation , the horizontal tangent line  is at .

The value of  is restricted to zero.

(b)

If the tangent line passes through the point .

Substitute  in the tangent line equation.

If  then point  and the tangent line is a horizontal tangent line.

If  then point .

Substitute  in the tangent line equation.

Restrictions for :

Here there are no restrictions for the value of .

The tangent line equation is .

 can be any real number.

(a) The tangent line equation is  and the value of  is restricted to zero.

(b) The tangent line equation is  and the value of  can be any real number.