JEE Advanced · 2008 · 13 Apr 2008 · Paper 2 · Circle

Consider

L_{1} : 2 x + 3 y + p - 3 = 0 L_{2} : 2 x + 3 y + p + 3 = 0,

where

p

is a real number, and

C : x^{2} + y^{2} + 6 x - 10 y + 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

(A)STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is a correct explanation for STATEMENT-1
(B)STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is NOT a correct explanation for STATEMENT-1
(C)STATEMENT-1 is True, STATEMENT-2 is False✓ Correct
(D)STATEMENT-1 is False, STATEMENT-2 is True

Answer: (C)

Solution

The circle

C

can be written as

(x + 3)^{2} + (y - 5)^{2} = 4

, with center

(- 3, 5)

and radius 2. The distance between parallel lines

L_{1}

and

L_{2}

\frac{6}{13} \approx 1.66 < 2

(the radius). So if

L_{1}

passes through the center (is a diameter),

L_{2}

is at distance

\frac{6}{13}

from the center, which is less than the radius, so

L_{2}

intersects the circle (is a chord). This makes STATEMENT-2 false. STATEMENT-1 is true because if

L_{1}

is a chord (not necessarily a diameter),

L_{2}

may or may not be a diameter depending on

p

JEE Advanced · 2008 · 13 Apr 2008 · Paper 2 · Single correct option