Consider the fourteen lines in the plane give | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(C) The given lines are $y=x+r$ and $y=-x+r$ for $r \in \{0, 1, 2, 3, 4, 5, 6\}$.These lines form a grid of squares.Two lines $y=x+r_1$ and $y=x+r_2$

Vedclass · Accepted Answer

$25$

Vedclass · Answer

(C) The given lines are $y=x+r$ and $y=-x+r$ for $r \in \{0, 1, 2, 3, 4, 5, 6\}$.These lines form a grid of squares.Two lines $y=x+r_1$ and $y=x+r_2$ are parallel,and the distance between them is $\frac{|r_1-r_2|}{\sqrt{1^2+(-1)^2}} = \frac{|r_1-r_2|}{\sqrt{2}}$.For the side length of the square to be $\sqrt{2}$,we require $\frac{|r_1-r_2|}{\sqrt{2}} = \sqrt{2}$,which implies $|r_1-r_2| = 2$.For the set $r \in \{0, 1, 2, 3, 4, 5, 6\}$,the pairs $(r_1, r_2)$ with a difference of $2$ are $(0, 2), (1, 3), (2, 4), (3, 5), (4, 6)$. There are $5$ such pairs.Similarly,for the lines $y=-x+r$,the distance between $y=-x+r_3$ and $y=-x+r_4$ is $\frac{|r_3-r_4|}{\sqrt{1^2+1^2}} = \frac{|r_3-r_4|}{\sqrt{2}}$.Setting this to $\sqrt{2}$ gives $|r_3-r_4| = 2$,which also yields $5$ pairs.The total number of squares formed is the product of the number of intervals of length $2$ in each direction,which is $5 	imes 5 = 25$.

Consider the fourteen lines in the plane given by $y=x+r$ and $y=-x+r$,where $r \in \{0, 1, 2, 3, 4, 5, 6\}$. The number of squares formed by these lines,whose sides are of length $\sqrt{2}$,is:

Similar Questions

$A$ straight line through the origin $O(0, 0)$ meets the lines $4x + 3y - 10 = 0$ and $8x + 6y + 5 = 0$ at points $A$ and $B$ respectively. Then $O$ divides the segment $AB$ in the ratio:

For $\lambda, \mu \in R$,$(x-2y-1)+\lambda(3x+2y-11)=0$ and $(3x+4y-11)+\mu(-x+2y-3)=0$ represent two families of lines. If the equation of the line common to both the families is $ax+by-5=0$,then $2a+b=$

$A$ straight line through the origin $O$ meets the lines $3y = 10 - 4x$ and $8x + 6y + 5 = 0$ at points $A$ and $B$ respectively. Then $O$ divides the segment $AB$ in the ratio:

The line joining two points $A(2,0)$ and $B(3,1)$ is rotated about $A$ in an anti-clockwise direction through an angle of $15^\circ$. The equation of the line in the new position is:

If the equation of the base of an equilateral triangle is $2x - y = 1$ and the vertex is $(-1, 2)$,then the length of the side of the triangle is

Vedclass Test Series

Exam Paper Generator

Online Exam Module