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JEE Advanced 2008Paper 2
Exam
JEE Advanced
Year
2008
Paper
Paper 2
Subject
Mathematics
Chapter
Straight Lines

JEE Advanced 2008 · Paper 2 · Mathematics · Straight Lines

Consider L_1: 2 x+3 y+p-3=0 ; L_2: 2 x+3 y+p+3=0 where p is a real...

Consider L1:2x+3y+p−3=0;L2:2x+3y+p+3=0L_1: 2 x+3 y+p-3=0 ; L_2: 2 x+3 y+p+3=0 where pp is a real number and C:x2+y2+6x−10y+30=0C: x^2+y^2+6 x-10 y+30=0. Statement 1 If line L1L_1 is a chord of circle CC, then line L2L_2 is not always a diameter of circle CC. Statement 2 If line L1L_1 is a diameter of circle CC, then line L2L_2 is not a chord of circle CC.

Options

  1. AStatement 1 is true, Statement 2 is true, Statement 2 is a correct explanation for Statement 1.
  2. BStatement 1 is true, Statement 2 is true, Statement 2 is not a correct explanation for Statement 1.
  3. CStatement 1 is true, Statement 2 is false.
  4. DStatement 1 is false, Statement 2 is true

Correct answer

C. Statement 1 is true, Statement 2 is false.

Step-by-step solution

Equation of circle CC is (x+3)2+(y−5)2=9+25−30=4⇒(x+3)2+(y−5)2=22 Centre =(3,−5)\begin{aligned} (x+3)^2+(y-5)^2 & =9+25-30=4 \\ \Rightarrow \quad \quad(x+3)^2+(y-5)^2 & =2^2 \\ \text{ Centre }& =(3,-5) \end{aligned}If L1L_1 is diameter, then 2(3)+3(−5)+p−3=0⇒p=12∴L12(3)+3(-5)+p-3=0 \Rightarrow p=12\therefore \quad L_1 is 2x+3y+9=0andL22 x+3 y+9=0 and L_2 is 2x+3y+15=02 x+3 y+15=0 Distance of centre of circle from L2=∣2(3)+3(−5)+1522+32∣=613<2L_2=\left|\frac{2(3)+3(-5)+15}{\sqrt{2^2+3^2}}\right|=\frac{6}{\sqrt{13}} \lt 2 \quad [radius of circle] ∴L2\therefore L_2 is a chord of circle $C. Statement 2 is false.
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