Consider where is a real number, and STATEMENT - 1: If line is a chord of circle , then line is not always a diameter of circle . STATEMENT - 2: If line is a diameter of circle , then line is not a chord of circle . [2008

Question

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 $C$ .

STATEMENT - 2: If line $L_{1}$ is a diameter of circle $C$ , then line $L_{2}$ is not a chord of circle $C$ . [2008]

Smart Achievers · Accepted Answer

(3)

Given: A circle $x^{2} + y^{2} + 6 x - 10 y + 30 = 0$ with centre $(- 3, 5)$ and radius = 2

$L_{1} : 2 x + 3 y + (p - 3) = 0$

$L_{2} : 2 x + 3 y + p + 3 = 0$

$Clearly L_{1} ∥ L_{2}$

$∴$ $Distance between L_{1} and L_{2}$

$=$ $| \frac{(p + 3) - (p - 3)}{\sqrt{2^{2} + 3^{2}}} |$ $= \frac{6}{\sqrt{13}} < 2$

$\Rightarrow$ If one line is a chord of the given circle, the other line may or may not be the diameter of the circle.

$∴$ Statement 1 is true and Statement 2 is false.

Want Us to Email you About Special Offers & Updates?