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