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.