Line Intersecting Circle at Two Points

Consider the equation of circle with centre at origin \left( {0,0}  \right) and radius r. Then equation of such a circle is written as

{x^2}  + {y^2} = {r^2}\,\,\,{\text{ -  -  - }}\left( {\text{i}} \right)

Now equation of a line is represented by

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

To find the point of intersection of line (ii) and the circle (i), using the method of solving simultaneous equation in which one equation is in quadratic and other is in linear form, so take value of y from equation (ii) and putting this value in equation (i) i.e. equation of circle as follows

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


Since equation (iii) is a quadratic equation in x, and we can solve this quadratic equation either by completing square method or using quadratic formula and can have at most roots i.e. values of x and putting values of x in equation (ii) to get the values of y and obtained two points. These points are the intersection of a line with circle, i.e. the line (ii) can intersect the circle (i) at most two points as clear from the given diagram.



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