Straight Lines question bank (Maths)

Let $A,B$ be points on the two half-lines $x-\sqrt{3}|y|=\alpha$, $\alpha\gt 0$ at a distance of $\alpha$ from their point of intersection $P$. The line segment $AB$ meets the angle bisector of the given half-lines at the point $Q$. If $PQ=\dfrac{9}{2}$ and $R$ is the radius of the circumcircle of $\triangle PAB$, then $\dfrac{\alpha^2}{R}$ is equal to ______

Let the line $x+y=1$ meet the axes of $x$ and $y$ at A and B, respectively. A right angled triangle AMN is inscribed in the triangle OAB , where O is the origin and the points M and N lie on the lines $O B$ and $A B$, respectively. If the area of the triangle $A M N$ is $\frac{4}{9}$ of the area of the triangle $O A B$ and AN : NB $=\lambda: 1$, then the sum of all possible value(s) of is $\lambda$ :