# Equation of the Tangent to the Conic

The equation of the tangent to the conic at the point can be written in the form

__Proof__**:**

Since the point lies on the conic

So the above equation (i) becomes

Now differentiating equation (i) of a general conic with respect to , we have

The equation of the tangent at the point is

Adding on both sides and using equation (ii), we have

__NOTE__**:** The above theorem suggests that the equation of the tangent to the conic at the point may be obtained by replacing by ; by ; by ; by ; by . Keeping these replacements in mind, we have the following conclusions:

**(i)** For parabola , the equation of the tangent is

**(ii)** For ellipse , the equation of the tangent is

**(iii)** For hyperbola , the equation of the tangent is