| Consider the family of straight lines \[2x{{\sin }^{2}}\theta +y{{\cos }^{2}}\theta =2\cos 2\theta \] |
| 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; 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\[6{{\sin }^{2}}\theta -2{{\cos }^{2}}\theta =2\cos 2\theta \] \[i.e.\] \[6(1-{{\cos }^{2}}\theta )-2{{\cos }^{2}}\theta =4{{\cos }^{2}}\theta -2\] \[i.e.\] \[12{{\cos }^{2}}\theta =8\] \[\therefore \]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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