Question

Write the condition that Ix + my + n = 0 may be a tangent to the circle x²+y² + 2gx + 2fy +C =0​

Answer 1

Answer:

$r= \frac{ |c - lg - fm| }{ \sqrt{ {l}^{2} + {m}^{2} } }$

Step-by-step explanation:

The given circle has equation

${x}^{2} + {y}^{2} + 2gx + 2fy + c = 0$

This circle has centre (-g,-f).

For the line

$lx + my + n = 0$

to be a tangent to this circle, the perpendicular distance from the center to this line must be equal to the radius of the circle.

This is given by:

$d = \frac{ |ax + by + c| }{ \sqrt{ {a}^{2} + {b}^{2} } }$

We substitute the center and radius to get:

$r=\frac{ |l \times - g + m \times - f+ c| }{ \sqrt{ {l}^{2} + {m}^{2} } }$

We simplify to get;

$r= \frac{ |c - lg - fm| }{ \sqrt{ {l}^{2} + {m}^{2} } }$