Vertices of a variable triangle are `(3,4); (5costheta, 5sintheta)` and `(5sintheta,-5costheta)` where `theta` is a parameter then the locus of its ci

Question

Vertices of a variable triangle are `(3,4); (5costheta, 5sintheta)` and `(5sintheta,-5costheta)` where `theta` is a parameter then the locus of its ci

Vertices of a variable triangle are `(3,4); (5costheta, 5sintheta)` and `(5sintheta,-5costheta)` where `theta` is a parameter then the locus of its circumcentre is
A. `(x+y-1)^2+(x-y-7)^2=100`
B. `(x+y-7)^2+(x-y-1)^2=100`
C. `(x+y-7)^2+(x+y-1)^2=100`
D. `(x+y-7)^2+(x-y+1)^2=100`

Monjur Alahi

5 views

2025-01-04 04:42

1 Answers

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

Correct Answer - D
Distance of all the points from (0,0) are 5 units. That means the circumcenter of the triangle formed by the given points is (0,0). If `G-=(h,k)` is the centroid of the triangle, then `3h=3+5(costheta+sintheta),3k=4+5(sintheta-costheta)`. If `H(alpha,beta)` is the orthocenter, then
`OG:GH=1:2or alpha=3h,beta=3k`
`costheta +sintheta =(alpha-3)/(5),sintheta-costheta=(beta-4)/(5)`
or `sintheta=(alpha+beta-7)/(10),costheta=(alpha-beta+1)/(10)`
Thus, the locus of `(alpha,beta)` is
`(x+y-7)^2+(x-y)+1)^2=100`