If equation `|sqrt((x-tantheta)^(2)+(y-sqrt3tantheta)^(2))-sqrt((x-2tantheta)^(2)+y^(2))|=2,theta in[0,pi]-{(pi)/(2)}` represents hyperbola, then find

Question

If equation `|sqrt((x-tantheta)^(2)+(y-sqrt3tantheta)^(2))-sqrt((x-2tantheta)^(2)+y^(2))|=2,theta in[0,pi]-{(pi)/(2)}` represents hyperbola, then find

If equation `|sqrt((x-tantheta)^(2)+(y-sqrt3tantheta)^(2))-sqrt((x-2tantheta)^(2)+y^(2))|=2,theta in[0,pi]-{(pi)/(2)}` represents hyperbola, then find the value of `theta`.

Monjur Alahi

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2025-01-04 04:42

1 Answers

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Salim Ahmed Kawsar · Accepted Answer · 2023-02-05T15:29:48+06:00

We have
`|underset("Distance between "F_(1) (tantheta,sqrt3tantheta)and P(x,y))(sqrt(ubrace((x-tantheta)^(2)+(y-sqrt3tantheta)^(2))))-underset("Distance between"F_(2)(2tan theta,0) and P(x,y))(sqrt(ubrace((x-2tantheta)^(2)+y^(2))))|=2`
So, `|R_(1)P-F_(2)P|=2,`
where `P-=(x,y),F_(1)-=(tan theta,sqrt3 tan theta) and F_(2)-=(2 tan theta, 0).`
Therefore, given equation represents a hyperbola if
`F_(1)F_(2)gt2`
`rArr" "sqrt((tantheta-2tan theta)^(2)+sqrt3 tan theta-0^(2))gt2`
`rArr" "|2tantheta|gt2`
`rArr" "|tan theta|gt1`
Hence, `thetain((pi)/(5),(pi)/(2))uu((pi)/(2),(3pi)/(4))`