# Equation of the Tangent to the Conic

The equation of the tangent to the conic $a{x^2} + b{y^2} + 2hxy + 2gx + 2fy + c = 0$ at the point $\left( {{x_1},{y_1}} \right)$ can be written in the form

Proof:
Since the point $\left( {{x_1},{y_1}} \right)$ lies on the conic

So the above equation (i) becomes

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

The equation of the tangent at the point $\left( {{x_1},{y_1}} \right)$ is

Adding $g{x_1} + f{y_1} + c$ 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 $\left( {{x_1},{y_1}} \right)$ may be obtained by replacing ${x^2}$ by $x{x_1}$; ${y^2}$ by $y{y_1}$; $2xy$ by $x{y_1} + {x_1}y$; $2x$ by $x + {x_1}$; $2y$ by $y + {y_1}$. Keeping these replacements in mind, we have the following conclusions:

(i) For parabola ${y^2} = 4ax$, the equation of the tangent is $y{y_1} = 2a\left( {x + {x_1}} \right)$

(ii) For ellipse $\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} = 1$, the equation of the tangent is $\frac{{x{x_1}}}{{{a^2}}} + \frac{{y{y_1}}}{{{b^2}}} = 1$

(iii) For hyperbola $\frac{{{x^2}}}{{{a^2}}} - \frac{{{y^2}}}{{{b^2}}} = 1$, the equation of the tangent is $\frac{{x{x_1}}}{{{a^2}}} - \frac{{y{y_1}}}{{{b^2}}} = 1$