A group of Class $X$ students goes to picnic during vacation, There were three different slides and three friends Ajay, Ram and Shyam are sliding in the three slides. The position of the three friends shown by $P, Q$ and $R$ in three different slides are given below:
.png)
Consider $\mathrm{O}$ as orgin, answer the below questions :
(i) The co-ordinates of the point ' $Q$ ' which divides the line segment PR in the ratio $1: 2$ internally:
.png)
(a) $\left(4, \frac{13}{3}\right)$
(b) ${ }^{\left(\frac{13}{3} \cdot \frac{11}{3}\right)}$
(c) $\left(\frac{10}{3} \cdot \frac{13}{3}\right)$
(d) ${ }^{\left(\frac{13}{3}, 4\right)}$
(ii) Find the distance PR:
(a) $2 \sqrt{10}$ units
(b) $\sqrt{36}$ units
(c) $\sqrt{38}$ units
(d) $\sqrt{20}$ units
(iii) The co-ordinates of point on $x$-axis which is at equal distance PQ is:
(a) $\left(\frac{11}{9}, 0\right)$
(b) $(3,0)$
(c) $\left(\frac{13}{9}, 0\right)$
(d) $(1,3)$
(iv) The coordinates of the mid-point of $P R$ is:
(a) $(-4,5)$
(b) $(4,5)$
(c) $(5,4)$
(d) $(4,4)$
(v) If we shift origin 'O' by 2 units towards right and 1 unit towards North. Then the co-ordinates of point is:
.png)
(a) $(2,6)$
(b) $(6,2)$
(c) $(-6,2)$
(d) $(-2,6)$
Question ID - 191898 | SaraNextGen Top Answer A group of Class $X$ students goes to picnic during vacation, There were three different slides and three friends Ajay, Ram and Shyam are sliding in the three slides. The position of the three friends shown by $P, Q$ and $R$ in three different slides are given below:
Consider $\mathrm{O}$ as orgin, answer the below questions :
(i) The co-ordinates of the point ' $Q$ ' which divides the line segment PR in the ratio $1: 2$ internally:
(a) $\left(4, \frac{13}{3}\right)$
(b) ${ }^{\left(\frac{13}{3} \cdot \frac{11}{3}\right)}$
(c) $\left(\frac{10}{3} \cdot \frac{13}{3}\right)$
(d) ${ }^{\left(\frac{13}{3}, 4\right)}$
(ii) Find the distance PR:
(a) $2 \sqrt{10}$ units
(b) $\sqrt{36}$ units
(c) $\sqrt{38}$ units
(d) $\sqrt{20}$ units
(iii) The co-ordinates of point on $x$-axis which is at equal distance PQ is:
(a) $\left(\frac{11}{9}, 0\right)$
(b) $(3,0)$
(c) $\left(\frac{13}{9}, 0\right)$
(d) $(1,3)$
(iv) The coordinates of the mid-point of $P R$ is:
(a) $(-4,5)$
(b) $(4,5)$
(c) $(5,4)$
(d) $(4,4)$
(v) If we shift origin 'O' by 2 units towards right and 1 unit towards North. Then the co-ordinates of point is:
(a) $(2,6)$
(b) $(6,2)$
(c) $(-6,2)$
(d) $(-2,6)$
A group of Class $X$ students goes to picnic during vacation, There were three different slides and three friends Ajay, Ram and Shyam are sliding in the three slides. The position of the three friends shown by $P, Q$ and $R$ in three different slides are given below:
Consider $\mathrm{O}$ as orgin, answer the below questions :
(i) The co-ordinates of the point ' $Q$ ' which divides the line segment PR in the ratio $1: 2$ internally:
(a) $\left(4, \frac{13}{3}\right)$
(b) ${ }^{\left(\frac{13}{3} \cdot \frac{11}{3}\right)}$
(c) $\left(\frac{10}{3} \cdot \frac{13}{3}\right)$
(d) ${ }^{\left(\frac{13}{3}, 4\right)}$
(ii) Find the distance PR:
(a) $2 \sqrt{10}$ units
(b) $\sqrt{36}$ units
(c) $\sqrt{38}$ units
(d) $\sqrt{20}$ units
(iii) The co-ordinates of point on $x$-axis which is at equal distance PQ is:
(a) $\left(\frac{11}{9}, 0\right)$
(b) $(3,0)$
(c) $\left(\frac{13}{9}, 0\right)$
(d) $(1,3)$
(iv) The coordinates of the mid-point of $P R$ is:
(a) $(-4,5)$
(b) $(4,5)$
(c) $(5,4)$
(d) $(4,4)$
(v) If we shift origin 'O' by 2 units towards right and 1 unit towards North. Then the co-ordinates of point is:
(a) $(2,6)$
(b) $(6,2)$
(c) $(-6,2)$
(d) $(-2,6)$
(i) (a) $\left(4, \frac{13}{3}\right)$
Explanation :
The co-ordinate of $P$ and $R$ are $(2,5)$ and $(8,3)$ is
$
\begin{aligned}
&X=\frac{m_{1} x_{2}+m_{2} x_{1}}{m_{1}+m_{2}}=\frac{1(8)+2(2)}{3} \\
&=\frac{12}{3}=4
\end{aligned}
$
$y=\frac{m_{1} y_{2}+m_{2} y_{1}}{m_{1}+m_{2}}=\frac{1(3)+2(5)}{3}=\frac{13}{3}$
(ii) (a) $2 \sqrt{10}$ units
Explanation :
$
\begin{aligned}
&P R=\sqrt{\left(x_{2}-x_{1}\right)^{2}+\left(y_{2}-y_{1}\right)^{2}} \\
&=\sqrt{(8-2)^{2}+(3-5)^{2}} \\
&=\sqrt{\left(2^{2}+22^{2}\right.}=\sqrt{40}=2 \sqrt{10} \text { units }
\end{aligned}
$
(iii) (c) $\left(\frac{13}{9}, 0\right)$
Explanation :
Let $\mathrm{A}(x, 0)$ be the point on $x$-axis which is equidistant from $P$ and $Q$, then we have $\mathrm{PA}=\mathrm{QA}$
The coordinate point of $\mathrm{Q}=\left(4, \frac{13}{3}\right)$
$
\begin{aligned}
&\Rightarrow \mathrm{PA}^{2}=\mathrm{QA}^{2} \\
&(2-x)^{2}+(5-0)^{2}=(4-x)^{2}+\left(\frac{13}{3}-0\right)^{2}
\end{aligned}
$
$
\begin{aligned}
&\Rightarrow 4+x^{2}-4 x+25=16+x^{2}-8 x+\frac{1 \oplus 9}{9} \\
&\Rightarrow 4 x=16+\frac{1 \oplus}{9}-4-25 \\
&\Rightarrow x=\frac{13}{9}
\end{aligned}
$
Hence the point $\left(\frac{13}{9}, 0\right)$.
(iv) (c) $(5,4)$
Explanation:
Mid-point of $\mathrm{PR}=\left(\frac{x_{1}+x_{2}}{2}, \frac{y_{1}+y_{2}}{2}\right)$
$
\begin{aligned}
&=\left(\frac{2+8}{2}, \frac{5+3}{2}\right) \\
&=\left(\frac{10}{2}, \frac{8}{2}\right) \\
&=(5,4)
\end{aligned}
$
(v) (b) $(6,2)$
Explanation :
The co-ordinates of point $\mathrm{R}$ is $(6,2)$.