Let $A$, $B$, $C$ be 3 points on the parabola $y^2=4x$. Let $D$, $E$, $F$ be midpoints of $AB$, $BC$ and $AC$ respectively. If $G$ is the centroid of $\triangle DEF$ and normals to the parabola at $A$, $B$ and $C$ are concurrent at $H(5,1)$, then slope of the line $GH$ is
Consider: I) General form $ax+by+c=0$ ($a,b,c\in\mathbb{R}$) always represents a straight line. II) A line $ax+by+1=0$ is such that algebraic sum of perpendiculars from $(x_i,y_i)$ $(i=1,2,\ldots,n)$ is zero, then line always passes through a fixed point $\left(\bar{x},\bar{y}\right)$ where $\bar{x}=\frac{\sum x_i}{n}$ and $\bar{y}=\frac{\sum y_i}{n}$.
Let A ∈ (0, 0), B ∈ (5, 0), C ∈ (5, 3) and D ∈ (0, 3) are the vertices of rectangle ABCD. If P is a variable point lying inside the rectangle ABCD and d(P, L) denotes perpendicular distance of point P from line L. If d(P, AB) ≤ min{d(P, BC), d(P, CD), d(P, AD)}, then area of the region in which P lies is :
Consider three curves $H:(x+a)y=\lambda$, $\lambda<0$, $a>0$; $C: x^2+y^2-21y+109=0$; $P: y^2=bx$, $b>0$. Let the line $2x-y+8=0$ touch $H$, $C$ and $P$ at $L$, $M$, $N$ respectively such that $LM=MN=\sqrt{45}$. Then $8\lambda+b+a$ is