JEE Advance - Mathematics (2008 - Paper 2 Offline - No. 15)
$$\,{L_1}:\,\,2x\,\, + \,\,3y\, + \,p\,\, - \,\,3 = 0$$
$$\,{L_2}:\,\,2x\,\, + \,\,3y\, + \,p\,\, + \,\,3 = 0$$
where p is a real number, and $$\,C:\,{x^2}\, + \,{y^2}\, + \,6x\, - 10y\, + \,30 = 0$$
STATEMENT-1 : If line $${L_1}$$ is a chord of circle C, then line $${L_2}$$ is not always a diameter of circle C
and
STATEMENT-2 : If line $${L_1}$$ is a diameter of circle C, then line $${L_2}$$ is not a chord of circle C.
Explanation
Equation of circle C is
$${(x + 3)^2} + {(y - 5)^2} = 9 + 25 - 30 = 4$$
$$ \Rightarrow {(x + 3)^2} + {(y - 5)^2} = {2^2}$$
Centre = (3, $$-$$5)
If L1 is diameter, then it passes through the center (3, $$-$$5),
$$ \therefore $$ $$2(3) + 3( - 5) + p - 3 = 0 \Rightarrow p = 12$$
$$\therefore$$ L1 is $$2x + 3y + 9 = 0$$
and L2 is $$2x + 3y + 15 = 0$$
Distance of centre of circle from $${L_2}(d) = \left| {{{2(3) + 3( - 5) + 15} \over {\sqrt {{2^2} + {3^2}} }}} \right| = {6 \over {\sqrt {12} }} < 2$$ [radius of circle]
$$\therefore$$ L2 is a chord of circle C.
Statement 2 is false.
Statement - 1 is true. If L1 is a chord, L2 may lie outside the circle.
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