Two straight lines rotate about two fixed points. If they start from their position of coincidence such that one rotates at the rate double that of the other. Then find the locus of their point of intersection of two straight lines

Explanation

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.