| Statement-1: All the lines of the given family pass through the point (3, 2). |
| Statement-2: All the lines of the given family pass through a fixed point. |
A) Statement-1 is false, Statement-2 is true
B) Statement-1 is true, Statement-2 is true, and Statement-2 is a correct explanation for Statement-1
C) Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.
D) Statement-1 is true, Statement-2 is false.
Correct Answer: A
Solution :
\[2{{\sin }^{2}}\theta x+{{\cos }^{2}}\theta y=2\cos 2\theta \] Statement-1: The line passes through the point (3, -2) If \[6\sin \theta -2{{\cos }^{2}}\theta =2\cos 2\theta \] i.e. \[6(1-{{\cos }^{2}}\theta )-2{{\cos }^{2}}\theta =4{{\cos }^{2}}\theta -2\] i.e. \[12{{\cos }^{2}}\theta =8\] \[\therefore \]Statement-1 is false. Statement: 2 \[(1-{{\cos }^{2}}\theta )x+{{\cos }^{2}}\theta y=4{{\cos }^{2}}\theta -2\] \[\therefore \]\[{{\cos }^{2}}\theta (-2x+y-4)+2x+2=0\] Family of lines passes through the point of intersection of line \[2x-y+4=0\]and \[x=-1\] \[\therefore \]The point is (-1,2) \[\therefore \]Statement-2 is true.
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