Question

Observe the following statements:
I. The lengths of the tangents from any point on the line $2x+3y=5$ to the circles $x^2+y^2=9$, and $x^2+y^2+4x+6y=19$ are equal in length.
II. There is only one point such that the tangents from it to the three given circles are equal in length. Then the correct statement is:

• A. Only I
• B. Only II
• C. Both I & II
• D. Neither I nor II

Answer 1

The correct option is C Both I & II

(1) $x^2+y^2=9$ and $x^2+y^2+4x+6y-19=0$
So, equation of radial axis to this circles is
$\Rightarrow 4x+6y-10=0$
$\Rightarrow 2x+3y=5$
So, from any point on $2x+3y-5=0$ (radial axis) length of tangents on this two given circles are equal.
II. If you have given three circle and we have to find locus of point from which we can draw equal length of tangents on given circles, then take radial axis of $s_1$ and $s_2$
$s_1-s_2=0$
This equation gives some line $L_1.$
and take radial axis of $s_1$ and $s_3$
$s_1-s_3=0$
This equation gives some line $L_2.$
By intersection of line $L_1$ and $L_2$, we get a radial centre which is the only point from which we can draw a equal length of tangents.
$\therefore$ Both I and II are True.