Question

In a triangle ABC, the sides \(b\) and \(c\) are the roots of the equation \(x^2-61x+820=0\) and \(A=\tan^{-1}\left(\frac{4}{3}\right)\), then \(a^2\) is equal to

• A. 1098
• B. 1096
• C. 1097
• D. 1095

Hint:

Use \(b^2+c^2=(b+c)^2-2bc\) to simplify cosine rule expressions quickly.

Answer 1

The Correct Option is C

Step 1: Use relation between roots and coefficients.
If \(b\) and \(c\) are roots of:
\[ x^2-61x+820=0 \]
Then:
\[ b+c = 61,\quad bc = 820 \]
Step 2: Find \(\cos A\).
\[ A=\tan^{-1}\left(\frac{4}{3}\right) \Rightarrow \tan A = \frac{4}{3} \]
Take right triangle with opposite = 4, adjacent = 3, hypotenuse = 5.
So:
\[ \cos A = \frac{3}{5} \]
Step 3: Apply cosine rule.
\[ a^2 = b^2 + c^2 - 2bc\cos A \]
Step 4: Compute \(b^2+c^2\).
\[ b^2+c^2 = (b+c)^2 - 2bc = 61^2 - 2(820) = 3721 - 1640 = 2081 \]
Step 5: Substitute values.
\[ a^2 = 2081 - 2(820)\left(\frac{3}{5}\right) \]
\[ = 2081 - 1640\left(\frac{3}{5}\right) = 2081 - 984 = 1097 \]
Final Answer:
\[ \boxed{1097} \]