Consider the lines 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 I: If the line L 1 is a c

Question

Consider the lines 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 I: If the line L 1 ​ is a chord of the circle C , then the line L 2 ​ is not always a diameter of the circle C . Statement II: If the line L 1 ​ is a diameter of the circle C , then the line L 2 ​ is not a chord of the circle C .

TestHub · Accepted Answer

Centre (-3, 5), radius 2; L 1 ​ ∥ L 2 ​ and the distance between them is 13 ​ 6 ​ < 2 . If L 1 ​ is a diameter, L 2 ​ is a parallel line at distance 13 ​ 6 ​ < 2 from the centre, so L 2 ​ is always a chord ⇒ Statement II is false. If L 1 ​ is a chord, L 2 ​ may or may not be a diameter, so L 2 ​ is not always a diameter ⇒ Statement I is true.

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