Three roots of the equation, x 4- px 3+ qx 2- rx + s =0 are tan A , tan B and tan C where A , B , C are the angles of a triangle. The fourth root of the biquadratic is :

Question

Three roots of the equation, x 4- px 3+ qx 2- rx + s =0 are tan A , tan B and tan C where A , B , C are the angles of a triangle. The fourth root of the biquadratic is : (A) (p-r/1-q+s) (B) (p-r/1+q-s) (C) (p+r/1-q+s) (D) (p+r/1+q-s)

Tardigrade Admin · Accepted Answer

Let the fourth root be tan D Now tan ( Sigma A)=( Sigma tan A- Sigma tan A tan B tan C/1- Sigma tan A tan B+Π tan A) tan D=(p-r/1-q+s)

Q. Three roots of the equation, $x^{4} - p x^{3} + q x^{2} - r x + s = 0$ are $tan A, tan B$ and $tan C$ where $A, B, C$ are the angles of a triangle. The fourth root of the biquadratic is :

$\frac{p - r}{1 - q + s}$

$\frac{p - r}{1 + q - s}$

$\frac{p + r}{1 - q + s}$

$\frac{p + r}{1 + q - s}$

Let the fourth root be tan D
Now $tan (Σ A) = \frac{Σ t a n A - Σ t a n A t a n B t a n C}{1 - Σ t a n A t a n B + Π t a n A} tan D = \frac{p - r}{1 - q + s}$

Q. Three roots of the equation, x4−px3+qx2−rx+s=0 are tanA,tanB and tanC where A,B,C are the angles of a triangle. The fourth root of the biquadratic is :

1−q+sp−r​

1+q−sp−r​

1−q+sp+r​

1+q−sp+r​