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Question

The difference between the two values of c is:


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Solution

Step- 1: Solve for the values of c:

We know that, for a triangle with angles A,B,C and sides a,b,c; cosA=b2+c2-a22bc

c2-2bccosA+b2-a2=0 is a quadratic equation in c

For a Quadratic equation px2+qx+r=0,

Sum of roots =-qp and product of roots =rp

Let c1,c2 be the roots of the quadratic equation in c and here p=1,q=-2bcosA,r=b2-a2

c1+c2=2bcosAc1·c2=b2-a2

Step- 2: Solve for the required value:

c1-c2=c1+c22-4c1·c2[p-q2=p+q2-4pq]=2bcosA2-4b2-a2=4b2cos2A-4b2+4a2=-4b21-cos2A+4a2=4a2-4b2sin2A[sin2θ+cos2θ=1]c1-c2=2a2-b2sin2A

Hence the difference between two values of c is 2a2-b2sin2A.


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