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
B
an ellipse
C
a hyperbola
D
None of these

Solution

## The correct option is A an ellipseLet 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.Maths

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