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

\[ax{x_1} + by{y_1} + h\left( {x{y_1} + {x_1}y} \right) + g\left( {x + {x_1}} \right) + f\left( {y + {y_1}} \right) + c = 0\]

Proof:
Since the point $$\left( {{x_1},{y_1}} \right)$$ lies on the conic
\[a{x^2} + b{y^2} + 2hxy + 2gx + 2fy + c = 0\,\,\,\,{\text{ – – – }}\left( {\text{i}} \right)\]

So the above equation (i) becomes
\[a{x_1}^2 + b{y_1}^2 + 2h{x_1}{y_1} + 2g{x_1} + 2f{y_1} + c = 0\,\,\,\,{\text{ – – – }}\left( {{\text{ii}}} \right)\]

Now differentiating equation (i) of a general conic with respect to $$x$$, we have
\[\begin{gathered} \frac{{dy}}{{dx}} = – \frac{{ax + hy + g}}{{hx + by + f}} \\ \Rightarrow {\left[ {\frac{{dy}}{{dx}}} \right]_{\left( {{x_1},{y_1}} \right)}} = – \frac{{a{x_1} + h{y_1} + g}}{{h{x_1} + b{y_1} + f}} \\ \end{gathered} \]

The equation of the tangent at the point $$\left( {{x_1},{y_1}} \right)$$ is
\[\begin{gathered} y – {y_1} = – \frac{{a{x_1} + h{y_1} + g}}{{h{x_1} + b{y_1} + f}}\left( {x – {x_1}} \right) \\ \Rightarrow ax{x_1} + by{y_1} + h\left( {x{y_1} + {x_1}y} \right) + gx + fy = a{x_1}^2 + b{y_1}^2 + 2h{x_1}{y_1} + g{x_1} + f{y_1} \\ \end{gathered} \]

Adding $$g{x_1} + f{y_1} + c$$ on both sides and using equation (ii), we have
\[ax{x_1} + by{y_1} + h\left( {x{y_1} + {x_1}y} \right) + g\left( {x + {x_1}} \right) + f\left( {y + {y_1}} \right) + c = 0\]

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 $$\frac{{x{x_1}}}{{{a^2}}} – \frac{{y{y_1}}}{{{b^2}}} = 1$$