Condition for Line Tangent to Ellipse

The condition for a line y = mx + c to be the tangent to the ellipse \frac{{{x^2}}}{{{a^2}}} +  \frac{{{y^2}}}{{{b^2}}} = 1is that c  =  \pm \sqrt {{a^2}{m^2} + {b^2}} and the tangent to the ellipse are y = mx \pm  \sqrt {{a^2}{m^2} + {b^2}} .
Consider the equation of a line is represented by

y =  mx + c\,\,\,{\text{ -  -  - }}\left( {\text{i}} \right)


Consider the standard equation of ellipse with vertex at origin \left( {0,0}  \right)can be written as

\frac{{{x^2}}}{{{a^2}}}  + \frac{{{y^2}}}{{{b^2}}} = 1\,\,\,{\text{ -   -  - }}\left( {{\text{ii}}}  \right)


To find the point of intersection of straight line (i) and the given ellipse (ii), using the method of solving simultaneous equation we solve equation (i) and equation (ii). Putting the value of y from equation (i) in equation (ii), we have

\begin{gathered} \frac{{{x^2}}}{{{a^2}}} + \frac{{{{\left( {mx  + c} \right)}^2}}}{{{b^2}}} = 1 \\ \Rightarrow \frac{{{b^2}{x^2} + {a^2}{{\left(  {mx + c} \right)}^2}}}{{{a^2}{b^2}}} = 1 \\ \Rightarrow {b^2}{x^2} + {a^2}{\left( {mx +  c} \right)^2} = {a^2}{b^2} \\ \Rightarrow {b^2}{x^2} + {a^2}\left(  {{m^2}{x^2} + 2mcx + {c^2}} \right) = {a^2}{b^2} \\ \Rightarrow {b^2}{x^2} + {a^2}{m^2}{x^2} +  2{a^2}mcx + {a^2}{c^2} - {a^2}{b^2} = 0 \\ \Rightarrow \left( {{a^2}{m^2} + {b^2}}  \right){x^2} + 2{a^2}mcx + {a^2}\left( {{c^2} - {b^2}} \right) = 0\,\,\,{\text{  -  -   - }}\left( {{\text{iii}}} \right) \\ \end{gathered}



condition-line-ellipse

If equation (iii) has equal roots, then the line equation (i) will intersect the ellipse (ii) at one point only and so is the tangent to the ellipse.
For equal roots, we have

\begin{gathered} {\text{Discriminant  =  0}} \\ \Rightarrow {\left( {2{a^2}mc} \right)^2} -  4\left( {{a^2}{m^2} + {b^2}} \right){a^2}\left( {{c^2} - {b^2}} \right) = 0 \\ \Rightarrow 4{a^4}{m^2}{c^2} - 4{a^2}\left(  {{a^2}{m^2} + {b^2}} \right)\left( {{c^2} - {b^2}} \right) = 0 \\ \Rightarrow {a^2}{m^2}{c^2} - \left(  {{a^2}{m^2} + {b^2}} \right)\left( {{c^2} - {b^2}} \right) = 0 \\ \Rightarrow   - {b^2}{c^2} + {a^2}{m^2}{b^2} + {b^4} = 0 \\ \Rightarrow {c^2} - {a^2}{m^2} - {b^2} = 0 \\ \Rightarrow {c^2} = {a^2}{m^2} + {b^2} \\ \Rightarrow c =  \pm \sqrt {{a^2}{m^2} + {b^2}} \\ \end{gathered}


Putting these values of c in the equation of straight line (i), we have

y =  mx \pm \sqrt {{a^2}{m^2} + {b^2}}


These are the tangents to the ellipse as shown in the given diagram

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