Tag: reflection w.r.t a line

Questions Related to reflection w.r.t a line

The image of the point (-5,4) under a reflection across the y-axis is (5,4).

maths position and movement reflection w.r.t a line transformation transformation and symmetry in geometrical shapes

  1. True

  2. False

  3. Ambiguous

  4. Data insufficient

Show answer
Correct Option: A

Image of $\left (1,2\right)\ w.r.t\left (-2,-1\right)$ is

maths position and movement reflection w.r.t a line transformation transformation and symmetry in geometrical shapes

  1. $\left (0,5\right)$

  2. $\left (-4,-3\right)$

  3. $\left (-4,-2\right)$

  4. $\left (-4,-5\right)$

Show answer
Correct Option: D

If the line $\left (2\cos \theta+ 3\sin \theta\right)$ $x+(\left (3\cos \theta- 5\sin \theta\right)$ $y-\left (5\cos \theta- 2\sin \theta\right)=0$ passes through a fixed point $P$ for all values $\theta$ and $Q$ be the image of the point $P$ with the respect to the line $4x+6y-23=0$, then the distance of $Q$ from the origin is:

maths position and movement reflection w.r.t a line transformation transformation and symmetry in geometrical shapes

  1. $\dfrac {13}{5}$

  2. $\sqrt {5}$

  3. $5\sqrt {2}$

  4. $5$

Show answer
Correct Option: A

The image of the point $A(1,2)$ by the line mirror  $y=x$ is the 
Point B and the image of B by the line mirror $y=0$ is the point $(a,\beta )then:$ 

maths position and movement reflection w.r.t a line transformation transformation and symmetry in geometrical shapes

  1. $a = -2,\beta = - 1$

  2. $a = 0,\beta = 0$

  3. $a = 2,\beta = - 1$

  4. none of these

Show answer
Correct Option: A

A ray of light along $x + \sqrt {3y}  = \sqrt 3 $ gets reflected upon reaching $x - axis$ , then equation of the reflected ray is 

maths position and movement reflection w.r.t a line transformation transformation and symmetry in geometrical shapes

  1. $y = x + \sqrt 3 $

  2. $\sqrt 3 y = x - \sqrt 3 $

  3. $y = \sqrt 3 x - \sqrt 3 $

  4. $\sqrt 3 y = x - 1$

Show answer
Correct Option: B
Explanation:
Mirror or reflecting surface/ boundary is $y = 0$ or $x-axis$.

Light Ray: $x + \sqrt3 y = \sqrt3$ or,
$\sqrt 3 y = \sqrt3 - x$.

Slope: $m = \dfrac{-1}{\sqrt3}$. Meets x axis at $A(\sqrt3, 0)$.

The reflected ray will have a slope$ = - m = \dfrac{1}{\sqrt3}$. Reason is that the angle of inclination with x axis becomes 180 - the angle of incident ray.

Also it passes through point $A$.

So the equation is: $y - 0 =\dfrac{ (x - \sqrt3)}{ \sqrt3}.$
Or, $\sqrt3 y - x + \sqrt3 = 0$.
Or, $\sqrt3 y=x-\sqrt3$

The line segment joining $A\left( {3,\,\,0} \right),\,\,B\left( {5,\,\,2} \right)$ is rotated about a point A in anticlockwise sense through an angle $\displaystyle{\pi  \over 4}$ and B move to C. If a point D be the reflection of C in y-axis, then D=

maths position and movement reflection w.r.t a line transformation transformation and symmetry in geometrical shapes

  1. $\left( { - 3,\,2\sqrt 2 } \right)$

  2. $\left( {3,\,2\sqrt 2 } \right)$

  3. $\left( {3,\, - 2\sqrt 2 } \right)$

  4. $\left( {3,\,8\sqrt 2 } \right)$

Show answer
Correct Option: A
Explanation:
$A(3, 0)$ and $B(5, 2)$
Slope of AB$=\dfrac{2-0}{5-3}=1$
Then, if $\theta$, is the angle made by AB, with positive direction of x-axis, we have $\tan\theta =1$
$\Rightarrow \theta =45^o$
Given, AB is rotated by $45^o$ to AC
Now AB$=\sqrt{(3-5)^2+(0-2)^2}=2\sqrt{2}$
So, coordinate of pr w$(3, 2\sqrt{2})$
Hence reflection of c in y-axis is $(-3, 2\sqrt{2})$.

The reflection of the point $(2, -1, 3)$ in the plane $3x-2y-z=9$ is?

maths position and movement reflection w.r.t a line transformation transformation and symmetry in geometrical shapes

  1. $\left(\dfrac{26}{7}, \dfrac{15}{7}, \dfrac{17}{7}\right)$

  2. $\left(\dfrac{26}{7}, \dfrac{-15}{7}, \dfrac{17}{7}\right)$

  3. $\left(\dfrac{16}{7}, \dfrac{26}{7}, \dfrac{-17}{7}\right)$

  4. $\left(\dfrac{1}{6}, \dfrac{2}{3}, \dfrac{3}{4}\right)$

Show answer
Correct Option: A

Find the image of the point $\displaystyle \left ( -2, -7 \right )$ under the transformation
$\left ( x, y \right )\rightarrow \left ( x-2y,-3x+y \right ).$

maths position and movement reflection w.r.t a line transformation transformation and symmetry in geometrical shapes

  1. $\displaystyle \left ( 12, -1 \right )$

  2. $\displaystyle \left ( -12, 1 \right )$

  3. $\displaystyle \left ( 2, 7 \right )$

  4. $\displaystyle \left ( -2, 7 \right )$

Show answer
Correct Option: A
Explanation:

Given transformation is 
$\left ( x, y \right )\rightarrow \left ( x-2y,-3x+y \right )$
So, the point $(-2,-7)$ $\rightarrow (-2-2(-7),-3(-2)-7)$
So, the image of the point $(-2,-7)$ under the given transformation is $ (12,-1)$

If $(-2, 6)$ is the image of the point $(4, 2)$ with respect to the line $L =$ $0$, then $L =$

maths position and movement reflection w.r.t a line transformation transformation and symmetry in geometrical shapes

  1. $6x - 4y -7 =0$

  2. $2x + 3y -5 =0$

  3. $3x - 2y + 5 =0$

  4. $3x - 2y + 10=0$

Show answer
Correct Option: C
Explanation:

Let points are $A(-2,6)$ & $B(4,2)$

$\therefore$    $L=0$ is $\bot $ to $AB$ & passes through the mid point slope $AB=\dfrac { -4 }{ 6 } =\dfrac { -2 }{ 3 } $
$\therefore$    slope of $L=0=3/2$
passes through $(1,4)$
$\therefore$    required equation is $\left( y-4 \right) =\frac { 3 }{ 2 } \left( x-1 \right) $
                                     $\Rightarrow 2y-8=3x-3$
                                     $\Rightarrow 3x-2y+5=0$        [C]

The image of the pair of lines represented by$\displaystyle :ax^{2}+2hxy+by^{2}= 0 $ by the line $ y= 0 $ mirror is:

maths position and movement reflection w.r.t a line transformation transformation and symmetry in geometrical shapes

  1. $\displaystyle :ax^{2}-2hxy-by^{2}= 0$

  2. $\displaystyle :bx^{2}-2hxy+ay^{2}= 0$

  3. $\displaystyle :bx^{2}+2hxy+ay^{2}= 0$

  4. $\displaystyle :ax^{2}-2hxy+by^{2}= 0$

Show answer
Correct Option: D
Explanation:

In ${ ax }^{ 2 }+2hxy+{ by }^{ 2 }=0 $ replacing $y$ by $-y$
we get, ${ ax }^{ 2 }+2hx\left( -y \right) +{ by }^{ 2 }=0 $
$\Rightarrow { ax }^{ 2 }-2hxy+{ by }^{ 2 }=0 $