Question 14 Marks
Read the following text carefully and answer the questions that follow:
A satellite image of a colony is shown below. In this view, a particular house is pointed out by a flag, which is situated at the point of intersection of $x$ and $y-$ axes. If we go $2 \ c$m east and $3 \ cm$ north from the house, then we reach to a Grocery store. If we go $4 \ c$m west and $6 \ cm$ south from the house, then we reach to an Electricians's shop. If we go $6 \ cm$ east and $8 \ c$m south from the house, then we reach to a food cart. If we go $6 \ cm$ west and $8 \ cm$ north from the house, then we reach a bus stand.
Scale:
$x-$ axis: $1 \ cm = 1$ unit
$y -$ axis: $1 \ cm = 1$ unit
$i.$ What is the distance between the grocery store and food cart?
$ii$. What is the distance of the bus stand from the house?
$iii$. If the grocery store and electricians shop lie on a line, then what will be the ratio of distance of house from grocery store to that from electrician's shop?
OR
What are the ratio of distances of the house from bus stand to food cart?
A satellite image of a colony is shown below. In this view, a particular house is pointed out by a flag, which is situated at the point of intersection of $x$ and $y-$ axes. If we go $2 \ c$m east and $3 \ cm$ north from the house, then we reach to a Grocery store. If we go $4 \ c$m west and $6 \ cm$ south from the house, then we reach to an Electricians's shop. If we go $6 \ cm$ east and $8 \ c$m south from the house, then we reach to a food cart. If we go $6 \ cm$ west and $8 \ cm$ north from the house, then we reach a bus stand.
Scale:
$x-$ axis: $1 \ cm = 1$ unit
$y -$ axis: $1 \ cm = 1$ unit
$i.$ What is the distance between the grocery store and food cart?
$ii$. What is the distance of the bus stand from the house?
$iii$. If the grocery store and electricians shop lie on a line, then what will be the ratio of distance of house from grocery store to that from electrician's shop?
OR
What are the ratio of distances of the house from bus stand to food cart?
Answer
View full question & answer→$i$. Consider the house is at origin $(0,0)$, then coordinates of grocery store, electrician's shop, food cart and bus stand are respectively $(2,3),(-4,-6),(6,-8)$ and $(-6,8)$.
Since, grocery store is at $(2,3)$ and food cart is at $(6,-8)$
$\therefore$ Required distance $=\sqrt{(6-2)^2+(-8-3)^2}$
$=\sqrt{4^2+11^2}=\sqrt{16+121}=\sqrt{137} \ cm$
$ii.$ Consider the house is at origin $(0,0)$, then coordinates of the grocery store, electrician's shop, food cart and bus stand are respectively $(2,3),(-4,-6),(6,-8)$ and $(-6,8)$.
Required distance
$=\sqrt{(-6)^2+8^2}=\sqrt{36+64}=\sqrt{100}=10 \ cm$
$iii$. Consider the house is at origin $(0,0)$, then coordinates of grocery store, electrician's shop, food cart and bus stand are respectively $(2,3),(-4,-6),(6,-8)$ and $(-6,8)$.
$0=\frac{2 k-4}{k+1}$
$\Rightarrow 2 k=4$
$\Rightarrow k=2$
Thus, $O$ divides $EG$ in the ratio $2: 1$
Hence, required ratio $= OG : OE$ i.e., $1: 2$.
OR
Consider the house is at origin $(0,0)$, then coordinates of grocery store, electrician's shop, food cart and bus stand are respectively $(2,3),(-4,-6),(6,-8)$ and $(-6,8)$.
Since, $(0,0)$ is the mid $-$ point of $(-6,8)$ and $(6,-8)$,
therefore both bus stand and food cart are at equal distances from the house. Hence, required ratio is $1: 1$.
Since, grocery store is at $(2,3)$ and food cart is at $(6,-8)$
$\therefore$ Required distance $=\sqrt{(6-2)^2+(-8-3)^2}$
$=\sqrt{4^2+11^2}=\sqrt{16+121}=\sqrt{137} \ cm$
$ii.$ Consider the house is at origin $(0,0)$, then coordinates of the grocery store, electrician's shop, food cart and bus stand are respectively $(2,3),(-4,-6),(6,-8)$ and $(-6,8)$.
Required distance
$=\sqrt{(-6)^2+8^2}=\sqrt{36+64}=\sqrt{100}=10 \ cm$
$iii$. Consider the house is at origin $(0,0)$, then coordinates of grocery store, electrician's shop, food cart and bus stand are respectively $(2,3),(-4,-6),(6,-8)$ and $(-6,8)$.
$0=\frac{2 k-4}{k+1}$
$\Rightarrow 2 k=4$
$\Rightarrow k=2$
Thus, $O$ divides $EG$ in the ratio $2: 1$
Hence, required ratio $= OG : OE$ i.e., $1: 2$.
OR
Consider the house is at origin $(0,0)$, then coordinates of grocery store, electrician's shop, food cart and bus stand are respectively $(2,3),(-4,-6),(6,-8)$ and $(-6,8)$.
Since, $(0,0)$ is the mid $-$ point of $(-6,8)$ and $(6,-8)$,
therefore both bus stand and food cart are at equal distances from the house. Hence, required ratio is $1: 1$.
