Question

Observe the following statements:
Assertion(A): The equation $x^2+y^2+2\lambda x+4=0$ represents a real circle for all $\lambda \in R$.
Reason (R ): The radical axis of any two circles of the family represented by $x^2+y^2+2\lambda x+4=0$ is the x-axis.
The correct statement among the following is:

• A. A is true, (R) is false
• B. (A) is false, (R) is true
• C. (A) is true, (R) is true
• D. (A) is false, (R) is false

Answer 1

The correct option is D (A) is false, (R) is false

Assertion (A): $x^2+y^2+2\lambda x+4=0$ represent real circle
then radius $=\sqrt{{\lambda }^2-4}\ge 0$
$\therefore \lambda ϵ(-\infty ,-2]\bigcup \left[\right(2,\infty ),$ False.
Reason(R): Family of circle represented by
$x^2+y^2+2\lambda x+4=0$
Let $s_1:x^2+y^2+2{\lambda }_{1}x+t=0$
$s_2:x^2+y^2+2{\lambda }_{2}x+4=0$
So, radical axis of this circles is
$s_1-s_2=0$
$({\lambda }_1-{\lambda }_2)x=0$
$x=0$
So, $Y-axis(x=0)$ is radical axis of given family of
circles.
(R) is also False.