Question

a triangle `A B C` with fixed base `B C` , the vertex `A` moves such that `cosB+cosC=4sin^2A/2dot` If `a ,ba n dc ,` denote the length of the sides of the triangle opposite to the angles `A , B ,a n dC` , respectively, then `b+c=4a` (b) `b+c=2a` the locus of point `A` is an ellipse the locus of point `A` is a pair of straight lines
A. `b+c=4a`
B. `b+c=2a`
C. locus of point A is an ellipse
D. locus of point A is a pair of straight line

Answer 1

Correct Answer - B::C
Given ` conB+cos =4 "sin"^(2) (A)/(2)`

` rArr 2 cos ((B+C)/(2))cos ((B-C)/(2))=4 "sin"^(2)(A)/(2)`
`2 "sin"(A)/(2)[ cos ((B-C)/(2))-2 "sin"(A)/2)]=0`
`rArr cos ((B-C)/(2))-2 cos ((B+C)/(2))= as "sin"(A)/(2)ne0`
`rArr-"cos"(B)/(2)"cos"(C)/(2)+3 "sin"(B)/(2)"sin"(C)/(2)=0`
`rArr "tan" (B)/(2) "tan"(C)/(2)=(1)/(3)`
`rArr sqrt(((s-a)(s-c))/(s(s-b))*((s-b)(s-a))/(s(s-c)))=(1)/(3)`
`rArr(s-a)/(s)=(1)/(3) rArr 2s=a`
`rArr b+c=2a`
`:.` Locus of A is an ellipse