IIT-JEE 2008 · Paper 2 · Mathematics Q12

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MathematicsAssertion and reason+3 / -1Question 12 of 66

Consider
$L_1:2x+3y+p-3=0,\qquad L_2:2x+3y+p+3=0,$
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 $C$ .

STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is a correct explanation for STATEMENT-1

STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is NOT a correct explanation for STATEMENT-1

STATEMENT-1 is True, STATEMENT-2 is False

STATEMENT-1 is False, STATEMENT-2 is True

Answer and worked solution

Official answer: C

Circle. Centre $(-3,5)$ , radius $\sqrt{9+25-30}=2$ .

Statement 1. $L_1$ and $L_2$ are parallel, $\frac6{\sqrt{13}}\approx1.66$ apart. If $L_1$ is a chord, $L_2$ passes through the centre only for one particular $p$ , so it is not always a diameter. True.

Statement 2. $L_1$ is a diameter when $-6+15+p-3=0$ , i.e. $p=-6$ . Then $L_2$ is at distance $\frac6{\sqrt{13}}<2$ from the centre, so it is a chord. False.

(C)

Topics:Equation of a circle Distance of a point from a line

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IIT-JEE 2008 · Paper 2, Mathematics question 12