### Possible intersection between a straight line and a circle

Question Sample Titled 'Possible intersection between a straight line and a circle'

 For the simultaneous equations

 ${y}={m}{x}+{c}$ ${x}^{{2}}+{y}^{{2}}+{D}{x}+{E}{y}+{F}={0}$

 by subsitituting the equations of the straight line ${y}={m}{x}+{c}$ into the equation of the circle ${x}^{{2}}+{y}^{{2}}+{D}{x}+{E}{y}+{F}={0}$ ,  we can obtain the quadratic equation ${x}^{{2}}+{\left({m}{x}+{c}\right)}^{{2}}+{D}{x}+{E}{\left({m}{x}+{c}\right)}+{F}={0}$ . By considering the discriminant ${\left(\Delta\right)}$ of this equation, we have the following three cases:

Discriminant${\left(\Delta={b}^{{2}}-{4}{a}{c}\right)}$$\Delta\gt{0}$$\Delta={0}$$\Delta\lt{0}$
No. of intersections

 Note: ${1}.$ If $\Delta={0}$, then the straight line is a tangent to the circle. ${2}.$ 2. If the straight line is a tangent to the circle, then $\Delta={0}$ .

 Example Determine the number of intersections between the straight line ${L}$: ${2}{x}+{3}{y}-{6}$ $={0}$ and the circle ${S}$: ${x}^{{2}}+{y}^{{2}}-{4}{x}+{2}{y}+{1}={0}$ .

 Solution ${2}{x}+{3}{y}-{6}$ $={0}$ $\ldots{\left({1}\right)}$ ${x}^{{2}}+{y}^{{2}}-{4}{x}+{2}{y}+{1}$ $={0}$ $\ldots{\left({2}\right)}$ From ${\left({1}\right)}$ , we have  ${y}$ $=-\dfrac{{2}}{{3}}{x}+{2}$ $\ldots{\left({3}\right)}$ By substituting ${\left({3}\right)}$ into ${\left({2}\right)}$ , we have  ${x}^{{2}}+{\left(-\dfrac{{2}}{{3}}{x}+{2}\right)}^{{2}}-{4}{x}+{2}{\left(-\dfrac{{2}}{{3}}{x}+{2}\right)}+{1}$ $={0}$ ${13}{x}^{{2}}-{72}{x}+{81}$ $={0}$ For the equation ${13}{x}^{{2}}-{72}{x}+{81}={0}$ ,  $\Delta={\left(-{72}\right)}^{{2}}-{4}{\left({13}\right)}{\left({81}\right)}={972}\gt{0}$ ∴   There are two intersections between the straight line and the circle.

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