MCQ
A variable line passes through a fixed point $P$. The algebraic sum of the perpendicular drawn from $(2,0)$, $(0, 2)$ and $(1, 1)$ on the line is zero, then the coordinates of the $P$ are
- A$(1,\, -1)$
- ✓$(1, \,1)$
- C$(2, \,1)$
- D$(2,\, 2)$
Now according to the condition
$\frac{{ - 2m + (m{x_1} - {y_1})}}{{\sqrt {1 + {m^2}} }} + \frac{{2 + (m{x_1} - {y_1})}}{{\sqrt {1 + {m^2}} }} + \frac{{1 - m + (m{x_1} - {y_1})}}{{\sqrt {1 + {m^2}} }} = 0$
==> $3 - 3m + 3m{x_1} - 3{y_1} = 0 \Rightarrow {y_1} - 1 = m({x_1} - 1)$
Since it is a variable line, so hold for every value of $m$. Therefore ${y_1} = 1,{x_1} = 1 \Rightarrow P(1,\,1)$.
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
| $\mathrm{x}$ | $-2$ | $-1$ | $3$ | $4$ | $6$ |
| $\mathrm{P}(\mathrm{X}=\mathrm{x})$ | $\frac{1}{5}$ | $\mathrm{a}$ | $\frac{1}{3}$ | $\frac{1}{5}$ | $\mathrm{~b}$ |
If the mean of $X$ is $2.3$ and variance of $X$ is $\sigma^{2}$, then $100 \sigma^{2}$ is equal to :