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Two rods are rotating about two fixed po...

Two rods are rotating about two fixed points in opposite directions. If they start from their position of coincidence and one rotates at the rate double that of the other, then find the locus of point of the intersection of the two rods.

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Suppose two rods are coincident on the x-aixs. One rotates about point O and the other about point A (a, 0).

If they rotate according to the question, then at some time t, they are in the position as shown in the figure. Form the figure.
`tan theta=(k)/(h) and tan2theta=(k)/(a-h)`
`"or "(2tantheta)/(1-tan^(2)theta)=(k)/(a-h)`
`"or "(2k//h)/(1-(k^(2)//h^(2)))=(k)/(a-h)`
`"or "(2hk)/(h^(2)-k^(2))=(k)/(a-h)`
`"or "2h(a-h)=h^(2)-k^(2)`
`"or "2ah-2h^(2)=h^(2)-k^(2)`
`"or "3x^(2)-y^(2)-2ax=0`
Therefore, the locus is a hyperbola.
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