Relation between Slopes and Coefficients in Equations of Pair of Lines

Q. Let $C_1:y=4x;C_2:y=\sin x\text{and}{C}_3:y=f\left(x\right)$ be the three curves passing through origin 'O' and defined in $[0,\frac{\pi }{2})$. From a point P on $C_2$ lines parallel to x-axis and y-axis meet $C_1\text{and}{C}_3$ at Q and R respectively as shown such that areas of curved regions OPQ and OPR are equal for all positions of point P. If $f\left(\frac{\pi }{4}\right)=\frac{1}{\sqrt{2}}+\frac{1}{a}-\frac{\pi }{b\sqrt{2}};$ (where a and b are integers) then (a + b) equal to

Q. Let in a $△ABC,$ $x,y,z$ are the lengths of perpendicular drawn from the vertices of the triangle to the opposite sides $a,b,c$ and $\cot A+\cot B+\cot C=k(\frac{1}{x^2}+\frac{1}{y^2}+\frac{1}{z^2}),$ then the value of $k$ is
(where $R,r,S,\Delta$ are circumradius, inradius, semiperimeter and area of triangle respectively.)

1. $R^2$
2. $rR$
3. $\Delta$
4. $a^2+b^2+c^2$

Q. Let $A$ and $B$ be any two points on the lines represented by $4x^2-9y^2=0.$ If the area of triangle $OAB$ is $5$ $\left(O\text{is origin}\right)$ then which of the following is the possible equation of the locus of midpoint of $AB?$

1. $4x^2+9y^2+30=0$
2. $4x^2-9y^2-30=0$
3. $9x^2-4y^2-30=0$
4. $9x^2+4y^2-30=0$

Q. Find the angle between the line $\overrightarrow{r}=(3,-2,4)+\lambda (2,2,1);\lambda \in R$ and the plane $2x-2y+z+7=0$.

Q. Consider a triangle $ABC$ and let $a,b,$ and $c$ denote the lenths of the sides opposite to vertices $A,B$ and $C$ respectively. Suppose $a=6,b=10$ and the area of the triangle is $15\sqrt{3}$. If $\angle ACB$ is obtuse and if $r$ dentoes the radius of the incricle of the triangle, then $r^2$ is equal to\

Q. If origin is centre of gravity of a triangle whose two vertices are are $(\cos \alpha ,\sin \alpha )$ and $(\cos \beta ,\sin \beta )$ then maximum area of such triangle is

1. $1$ sq. units
2. $1.5$ sq. units
3. $2$ sq. units
4. $2.5$ sq. units

Q. Let $A$ and $B$ be any two points on the lines represented by $4x^2-9y^2=0.$ If the area of triangle $OAB$ is $5$ $\left(O\text{is origin}\right)$ then which of the following is the possible equation of the locus of midpoint of $AB?$

1. $4x^2+9y^2+30=0$
2. $4x^2-9y^2-30=0$
3. $9x^2-4y^2-30=0$
4. $9x^2+4y^2-30=0$