Question

the line ax+by+c=0 touches the circle x²+y²=r². find the condition.
kindly solve and tell please​

Answer 1

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Answer 2

Answer:

The condition for line to touch the circle is $r=\frac{|c|}{\sqrt{a^{2}+b^{2}}}$

Step-by-step explanation:

Given that,

The line $ax+by+c=0$ touches the circle $x^{2} +y^{2}=r^{2}$ and we are required to find the condition for this to happen.

For the line to touch the circle,the perpendicular distance from center of circle to the line must be equal to the radius of the circle.

The perpendicular distance of a line from point $(x_{1},y_{1})$ is

$d=\frac{|ax_{1}+by_{1}+c|}{\sqrt{a^{2}+b^{2}} }$

The center of the given circle is (0,0) hence the perpendicular distance is

$\frac{|0+0+c|}{\sqrt{a^{2}+b^{2}} }=\frac{|c|}{\sqrt{a^{2}+b^{2}}}$

This distance is equal to the radius of the circle.Then

$r=\frac{|c|}{\sqrt{a^{2}+b^{2}}}$

Therefore,

The condition for line to touch the circle is $r=\frac{|c|}{\sqrt{a^{2}+b^{2}}}$

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