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Question

If the extremities of a line segment of length $$l$$ move in two fixed perpendicular straight lines, then the locus of that point which divides this lines segment in the ratio $$1:2$$ is


A
a parabola
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B
an ellipse
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C
a hyperbola
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D
None of these
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Solution

The correct option is A an ellipse
Let the two fixed $$\bot$$ straight lines be the coordinates axes.
Let $$P(h,k)$$ be the point whose locus is required.
Let $$PA:PB=1:2$$
Then $$\displaystyle PA=\frac { l }{ 3 } $$ and $$\displaystyle PB=\frac { 2l }{ 3 } $$
$$\displaystyle k=\frac { l }{ 3 } \sin { \theta  } \Rightarrow 3k=l\sin { \theta  } $$   ...(1)
and $$\displaystyle h=\frac { 2l }{ 3 } \cos { \theta  } \Rightarrow \frac { 3h }{ 2 } =l\cos { \theta  } $$   ...(2)
Squaring and adding (1) and (2), we get
$$\displaystyle 9{ k }^{ 2 }+\frac { 9{ h }^{ 2 } }{ 4 } ={ l }^{ 2 }\Rightarrow 9{ h }^{ 2 }+36{ k }^{ 2 }=4{ l }^{ 2 }$$
$$\therefore$$ Locus of $$P\left( h,k \right) $$ is $$9{ x }^{ 2 }+36{ y }^{ 2 }=4{ l }^{ 2 }$$, which is an ellipse.

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