A straight line L passes through the point of intersection of the lines $x - y + 1 = 0$ and $2x + y - 7 = 0$. If L intersects the positive x-axis at $A(a, 0)$ and the positive y-axis at $B(0, b)$, then the minimum area of the triangle $OAB$ (where $O$ is the origin) is ....

If the combined equation of angle bisectors of the lines $x^2 - 2pxy - y^2 = 0$ is $x^2 - 2qxy - y^2 = 0$, then which of the following is true?

A tangent having slope $-\dfrac{1}{2}$ to the ellipse $3x^2 + 4y^2 = 12$ intersects the X-axis and Y-axis at the points A and B respectively. if O is the origin, then the area of $\triangle AOB$ is ...

If the function $f(x) = \dfrac{2\sqrt{2} - (\cos x + \sin x)^3}{1 - \sin 2x}$ is continuous at $x = \dfrac{\pi}{4}$, then the value of $f\left(\dfrac{\pi}{4}\right)$ is ...