THE POSITION OF A LINE WITH RESPECT TO A CIRCLE
`text(Method - I :)`
Let the equation of the circle be
`x^ 2 + y^2 = a^2` ............................................. `(i)`
and the equation of the line be
`y = mx +c`...........................................................`(ii)`
From (1) and (2)
`x^2 + (mx +c)^2 = a^2`
`x^2 ( 1 + m^2) + 2 cmx +c^2 - a^2 = 0` ............. `(iii)`
`text( Case-I :) `
When points of intersection are real and distinct, then equation `(iii)` has two distinct roots.
Discriminant `> 0`
`4m^2c^2 - 4(1 + m^2)(c^2 - a^2) > 0`
`a^2 > c^2/(1+m^2)`
`a> |c|/sqrt(1+m^2)=` length of perpendicular from `(0, 0)` to `y = mx + c`
Thus, a line intersects a given circle at two distinct points if radius of circle is greater than the length of perpendicular from centre of the circle to the line.
`text(Case-II :)`
When the points of intersection are coincident, the equation `(iii)` has two equal root
`D =0`
`=>a= |c|/(sqrt(1+m^2))`
`a=` length of the perpendicular from the point `(0, 0)` to `y = mx + c`
Thus, a line touches the circle if radius of circle is equal to the length of perpendicular from centre of the circle to the line or called 'CONDITION OF TANGENCY'.
`text(Case-III :)`
When the points of intersection are imaginary. In this case (iii ) has imaginary roots
`D < 0`
`a < |c|/sqrt(1+m^2) `
or `a <` length of perpendicular from `(0, 0)` to `y = mx + c`
Thus a line does not intersect a circle if the radius of circle is less than the length of perpendicular from centre of the circle to the line.
`text(Method - II :)`
Let `S = x^2 + y^2 + 2gx + 2fy + c = 0` be a circle and
`L = ax +by + c = 0` be a line.
Let `r` be the radius of the circle and `p` be the length of the perpendicular drawn from the centre `(- g, - f)` on the line `L`.
Then it can be seen easily from the figure that. If
`(1) p < r=>` the line intersects the circle in two distinct points.
`(2) p = r =>` the line touches the circle, i.e. the line is a tangent to the circle.
`(3) p > r =>` the line neither intersects nor touches the circle i.e. , passes outside the circle.
`(4) p = 0 =>` the line passes through the centre of the circle.
Let the equation of the circle be
`x^ 2 + y^2 = a^2` ............................................. `(i)`
and the equation of the line be
`y = mx +c`...........................................................`(ii)`
From (1) and (2)
`x^2 + (mx +c)^2 = a^2`
`x^2 ( 1 + m^2) + 2 cmx +c^2 - a^2 = 0` ............. `(iii)`
`text( Case-I :) `
When points of intersection are real and distinct, then equation `(iii)` has two distinct roots.
Discriminant `> 0`
`4m^2c^2 - 4(1 + m^2)(c^2 - a^2) > 0`
`a^2 > c^2/(1+m^2)`
`a> |c|/sqrt(1+m^2)=` length of perpendicular from `(0, 0)` to `y = mx + c`
Thus, a line intersects a given circle at two distinct points if radius of circle is greater than the length of perpendicular from centre of the circle to the line.
`text(Case-II :)`
When the points of intersection are coincident, the equation `(iii)` has two equal root
`D =0`
`=>a= |c|/(sqrt(1+m^2))`
`a=` length of the perpendicular from the point `(0, 0)` to `y = mx + c`
Thus, a line touches the circle if radius of circle is equal to the length of perpendicular from centre of the circle to the line or called 'CONDITION OF TANGENCY'.
`text(Case-III :)`
When the points of intersection are imaginary. In this case (iii ) has imaginary roots
`D < 0`
`a < |c|/sqrt(1+m^2) `
or `a <` length of perpendicular from `(0, 0)` to `y = mx + c`
Thus a line does not intersect a circle if the radius of circle is less than the length of perpendicular from centre of the circle to the line.
`text(Method - II :)`
Let `S = x^2 + y^2 + 2gx + 2fy + c = 0` be a circle and
`L = ax +by + c = 0` be a line.
Let `r` be the radius of the circle and `p` be the length of the perpendicular drawn from the centre `(- g, - f)` on the line `L`.
Then it can be seen easily from the figure that. If
`(1) p < r=>` the line intersects the circle in two distinct points.
`(2) p = r =>` the line touches the circle, i.e. the line is a tangent to the circle.
`(3) p > r =>` the line neither intersects nor touches the circle i.e. , passes outside the circle.
`(4) p = 0 =>` the line passes through the centre of the circle.