\text{If}\frac{\sin \theta +\cos \theta }{\sin \theta -4\cos \theta }=11,\text{then the value of}\frac{13{\sin }^2\theta +3{\cos }^2\theta -3}{{\cos }^2\theta }\text{is:}

The correct option is 405/2.

Step 1: Simplify the given trigonometric equation to find tan θ

We are given:

\frac{\sin \theta +\cos \theta }{\sin \theta -4\cos \theta }=11

Cross-multiplying the terms:

\sin \theta +\cos \theta =11(\sin \theta -4\cos \theta )

Expanding the right side:

\sin \theta +\cos \theta =11\sin \theta -44\cos \theta

Grouping like terms together:

45\cos \theta =10\sin \theta

Dividing both sides by 10\cos \theta:

\tan \theta =\frac{\sin \theta }{\cos \theta }=\frac{45}{10}=\frac{9}{2}

Step 2: Simplify the required expression in terms of tan θ

We need to find the value of:

\frac{13{\sin }^2\theta +3{\cos }^2\theta -3}{{\cos }^2\theta }

Dividing each term in the numerator by {\cos }^2\theta:

=13\frac{{\sin }^2\theta }{{\cos }^2\theta }+3\frac{{\cos }^2\theta }{{\cos }^2\theta }-\frac{3}{{\cos }^2\theta }

Using the identity \frac{1}{{\cos }^2\theta }={\sec }^2\theta =1+{\tan }^2\theta:

=13{\tan }^2\theta +3-3(1+{\tan }^2\theta )

=13{\tan }^2\theta +3-3-3{\tan }^2\theta

=10{\tan }^2\theta

Step 3: Substitute the value of tan θ

Substitute \tan \theta =\frac{9}{2} into the expression:

=10\times {\left(\frac{9}{2}\right)}^2

=10\times \frac{81}{4}

=\frac{405}{2}

Thus, the value of the given expression is 405/2.