Exercise-3

The lines $p\left(p^2+1\right)x-y+q=0$ and ${\left(p^2+1\right)}^{2}x+\left(p^2+1\right)y+2q=0$ are perpendicular to a common line for

The line $x+y=\left|a\right|$ and $ax-y=1$ intersect each other in the first quadrant. Then, the set of all possible values of $a$ in the interval

If $A\left(2,-3\right)$ and $B\left(-2,1\right)$ are two vertices of a triangle and third vertex moves on the line $2 x+3 y=9,$ then the locus of the centroid of the triangle is

If the line $2 x+y=k$ passes through the point which divides the line segment joining the points $(1,1)$ and $(2,4)$ in the ratio $3: 2,$ then $K$ equals

A ray of light along $x+\sqrt{3}y=\sqrt{3}$ gets reflected upon reaching $X-$axis, the equation of the reflected ray is

The $x$ -coordinates of the incentre of the triangle that has the coordinates of midpoints of its sides as$(0,1),$ $(1,1)$ and $(1,0)$ is

Let $a,b,c$ and $d$ be non-zero numbers. If the point of intersection of the lines $4ax+2ay+c=0$ and $5bx+2by+d=0$, lies in the fourth quadrant and is equidistant from the two axes, then

The number of points having both coordinates as integers that lie in the interior of the triangle with vertices $(0,0), (0,41)$ and $(41,0),$ is