Circles
6.0 Intersection of a line and a circle
6.0 Intersection of a line and a circle
Consider the standard form of circle ${x^2} + {y^2} = {a^2}$ and the equation of line be $y = mx + c$ which intersects the given circle. Put the value of $y$ from equation of line in standard form of circle, we get $${x^2} + {\left( {mx + c} \right)^2} = {a^2}$$ $$\left( {1 + {m^2}} \right){x^2} + 2mcx + {c^2} - {a^2} = 0...(1)$$
CASE I: When points of intersection of line and circle are real and distinct, then equation $(1)$ has two distinct roots, hence ${B^2} - 4AC > 0$.
$$4{m^2}{c^2} - 4\left( {1 + {m^2}} \right)\left( {{c^2} - {a^2}} \right) > 0$$ $${a^2} > \frac{{{c^2}}}{{1 + {m^2}}}$$ $$ a> \frac{{\left| c \right|}}{{\sqrt {1 + {m^2}} }} $$
$\frac{{\left| c \right|}}{{\sqrt {1 + {m^2}} }}$=length of perpendicular from $(0,0)$ to $y=mx+c$
We can conclude that a line intersects a given circle at two distinct points if radius of circle is greater than the length of perpendicular drawn from centre of circle to the line.
CASE II: Condition of Tangency
When the points of intersection are coincident, then equation $1$ has two equal roots, hence ${B^2} - 4AC = 0$.
$$4{m^2}{c^2} - 4\left( {1 + {m^2}} \right)\left( {{c^2} - {a^2}} \right) = 0$$ $${a^2} = \frac{{{c^2}}}{{1 + {m^2}}}$$
$\frac{{\left| c \right|}}{{\sqrt {1 + {m^2}} }} = $length of perpendicular from $(0,0)$ to $y=mx+c$
We can conclude that a line touches a given circle if radius of circle is equal to the length of perpendicular drawn from centre of circle to the line.
CASE III: When points of intersection are imaginary, then equation $(1)$ has imaginary roots, hence ${B^2} - 4AC < 0$.
$$4{m^2}{c^2} - 4\left( {1 + {m^2}} \right)\left( {{c^2} - {a^2}} \right) < 0$$ $${a^2} < \frac{{{c^2}}}{{1 + {m^2}}}$$ $$a\left( {radius} \right) < \frac{{\left| c \right|}}{{\sqrt {1 + {m^2}} }}$$
$\frac{{\left| c \right|}}{{\sqrt {1 + {m^2}} }} = $length of perpendicular from $(0,0)$ to $y=mx+c$
We can conclude that a line does not intersect a given circle if radius of circle is less than the length of perpendicular drawn from centre of circle to the line.
