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Straight Lines and Pair of Straight Lines question

2017 · 9 Apr · Shift 1 · Q43
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Straight Lines and Pair of Straight Lines question

2017 · 9 Apr · Shift 1 · Q43

JEE MainMathematicsStraight Lines and Pair of Straight LinesMCQ+4 / −1
A square, of each side 2, lies above the x-axis and has one vertex at the origin. If one of the sides passing through the origin makes an angle 30o with the positive direction of the x-axis, then the sum of the x-coordinates of the vertices of the square is :
  1. A
    23−12\sqrt 3 - 123​−1
  2. B
    23−22\sqrt 3 - 223​−2
  3. C
    3−2\sqrt 3 - 23​−2
  4. D
    3−1\sqrt 3 - 13​−1
View written solutionFree

Correct answer: B

  1. Set up the square

Let the vertex at the origin be O=(0,0)O=(0,0)O=(0,0).

One side of the square through the origin has length 222 and makes an angle 30∘30^\circ30∘ with the positive xxx-axis.

So the adjacent vertex on this side is

A=(2cos⁡30∘,2sin⁡30∘)=(2⋅32,2⋅12)=(3,1).A=(2\cos 30^\circ, 2\sin 30^\circ)=\left(2\cdot \frac{\sqrt3}{2}, 2\cdot \frac12\right)=(\sqrt3,1).A=(2cos30∘,2sin30∘)=(2⋅23​​,2⋅21​)=(3​,1).
  1. Find the direction of the other side through the origin

Since adjacent sides of a square are perpendicular, the other side from the origin must make angle 30∘+90∘=120∘30^\circ+90^\circ=120^\circ30∘+90∘=120∘ with the positive xxx-axis.

This choice is correct because the square lies above the xxx-axis.

Hence the second adjacent vertex is

B=(2cos⁡120∘,2sin⁡120∘)=(2⋅(−12),2⋅32)=(−1,3).B=(2\cos 120^\circ, 2\sin 120^\circ)=\left(2\cdot \left(-\frac12\right), 2\cdot \frac{\sqrt3}{2}\right)=(-1,\sqrt3).B=(2cos120∘,2sin120∘)=(2⋅(−21​),2⋅23​​)=(−1,3​).
  1. Find the fourth vertex

The fourth vertex is

C=A+B=(3,1)+(−1,3)=(3−1, 1+3).C=A+B=(\sqrt3,1)+(-1,\sqrt3)=(\sqrt3-1,\,1+\sqrt3).C=A+B=(3​,1)+(−1,3​)=(3​−1,1+3​).
  1. List all vertices and add their xxx-coordinates

The four vertices are:

O=(0,0),A=(3,1),B=(−1,3),C=(3−1,1+3).O=(0,0),\quad A=(\sqrt3,1),\quad B=(-1,\sqrt3),\quad C=(\sqrt3-1,1+\sqrt3).O=(0,0),A=(3​,1),B=(−1,3​),C=(3​−1,1+3​).

Their xxx-coordinates are

0,3,−1,3−1.0,\quad \sqrt3,\quad -1,\quad \sqrt3-1.0,3​,−1,3​−1.

So the sum is

0+3+(−1)+(3−1)=23−2.0+\sqrt3+(-1)+(\sqrt3-1)=2\sqrt3-2.0+3​+(−1)+(3​−1)=23​−2.
  1. Match with the options
23−22\sqrt3-223​−2

which is Option B.

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