Multiple choice

$\Delta$ ABC is not right-angled and is inscribed in a fixed circle. If $a, A, b, B$ be slightly varied keeping $c, C$ fixed, then $\dfrac {da}{cos A}+\dfrac {db}{cos B}=?$

  1. $2R$
  2. $\pi$
  3. $0$

  4. none of these
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

In a triangle, a/sinA = b/sinB = 2R. Differentiating with respect to the variables, the sum of the variations da/cosA + db/cosB relates to the properties of the circumcircle and the fixed side c, which sums to zero due to the geometric constraints.

AI explanation

Using the extended law of sines, a equals 2R sin A and b equals 2R sin B. Differentiating these while keeping the circumradius R and side c constant gives da equaling 2R cos A dA and db equaling 2R cos B dB, which transforms the expression into 2R(dA + dB). Since the angles of the triangle must sum to pi radians and angle C is fixed, varying A and B requires dA + dB to equal 0. Substituting this into the expression yields 2R times 0, resulting in 0.

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