Question

State whether the given statement is True or False. The line $x+3y=0$ is a diameter of the circle $x^2+y^2+6x+2y=0$

Answer 1

We have circle $x^2+y^2+6x+2y=0\dots \left(1\right)$
General equation of circle:
$x^2+y^2+2gx+2fy+c=0\dots \left(2\right)$
Comparing the equations $\left(1\right)$ and $\left(2\right)$ we get
$g=3,f=1,c=0$
$\Rightarrow$ Centre of given circle $C(-3,-1)$
[$\because$ Centre of circle is $(-g,-f)$ ]
Given line $x+3y=0$
Checking centre satisfy the equation of above line or not.
$\Rightarrow (-3)+3\times (-1)=0$
$\Rightarrow -6=0$ [ Which is false]
$\Rightarrow$ Given line is not a diameter of given circle
Hence, the given statement is false.