JEE MAIN - Mathematics (2011 - No. 22)
Statement - 1 : The point $$A(1,0,7)$$ is the mirror image of the point
$$B(1,6,3)$$ in the line : $${x \over 1} = {{y - 1} \over 2} = {{z - 2} \over 3}$$
Statement - 2 : The line $${x \over 1} = {{y - 1} \over 2} = {{z - 2} \over 3}$$ bisects the line
segment joining $$A(1,0,7)$$ and $$B(1, 6, 3)$$
$$B(1,6,3)$$ in the line : $${x \over 1} = {{y - 1} \over 2} = {{z - 2} \over 3}$$
Statement - 2 : The line $${x \over 1} = {{y - 1} \over 2} = {{z - 2} \over 3}$$ bisects the line
segment joining $$A(1,0,7)$$ and $$B(1, 6, 3)$$
Statement -1 is true, Statement -2 is true; Statement -2 is not a correct explanation for Statement -1.
Statement -1 is true, Statement - 2 is false.
Statement - 1 is false , Statement -2 is true.
Statement -1 is true, Statement -2 is true; Statement -2 is a correct explanation for Statement -1.
Explanation
The directions ratios of the line segment joining points
$$A\left( {1,0,7} \right)\,\,$$ and $$\,\,\,B\left( {1,6,3} \right)$$ are $$0,6, - 4.$$
The direction ratios of the given line are $$1,2,3.$$
Clearly $$1 \times 0 + 2 \times 6 + 3 \times \left( { - 4} \right) = 0$$
So, the given line is perpendicular to line $$AB.$$
Also, the mid point of $$A$$ and $$B$$ is $$\left( {1,3,5} \right)$$ which lies on the given line.
So, the image of $$B$$ in the given line is $$A$$, because the given line is the perpendicular bisector of line segment joining points $$A$$ and $$B$$, But statement -$$2$$ is not a correct explanation for statement$$-1.$$
$$A\left( {1,0,7} \right)\,\,$$ and $$\,\,\,B\left( {1,6,3} \right)$$ are $$0,6, - 4.$$
The direction ratios of the given line are $$1,2,3.$$
Clearly $$1 \times 0 + 2 \times 6 + 3 \times \left( { - 4} \right) = 0$$
So, the given line is perpendicular to line $$AB.$$
Also, the mid point of $$A$$ and $$B$$ is $$\left( {1,3,5} \right)$$ which lies on the given line.
So, the image of $$B$$ in the given line is $$A$$, because the given line is the perpendicular bisector of line segment joining points $$A$$ and $$B$$, But statement -$$2$$ is not a correct explanation for statement$$-1.$$
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