Tag: transformation and symmetry in geometrical shapes

Quizzes (2) Questions (65)

Questions Related to transformation and symmetry in geometrical shapes

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

If $B$ is reflection of $A(a,5)$ about line $4x-3y=0$, then area of triangle $ABC$ is equal to

  1. $\dfrac{253}{50}$
  2. $\dfrac{506}{25}$
  3. $\dfrac{253}{25}$
  4. $\dfrac{506}{50}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The reflection B of A(a, 5) across 4x-3y=0 is found using the reflection formula. The area of triangle ABC (where C is the origin or a fixed point) is calculated using the coordinates of A, B, and the intersection point.

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

Locus of the image of the point (2, 3) in the line (2x - 3y + 4) + k(x - 2y + 3) = 0, k $\in $ R, is a 

  1. straight line parallel to x-axis

  2. straight line parallel to y-axis
  3. Circle of radius $\sqrt { 2 } $

  4. circle of radius 3
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The family of lines (2x-3y+4) + k(x-2y+3) = 0 passes through a fixed point (intersection of the two lines). The locus of the image of a point reflected across a family of lines passing through a fixed point is a circle centered at that fixed point.

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

The distance of the image of a point (or an object) from the line of symmetry (mirror) is  ----- as that of the point (object )from the line (mirror).

  1. same

  2. double

  3. triple

  4. none
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation
To make the above statement true, the word to be placed in the blank is : “same”
So, the true statement becomes :
The distance of the image of a point (or an object) from the line of symmetry (mirror) is same as that of the point (object) from the line (mirror).
Hence, option A is the correct answer.
Multiple choice maths position and movement reflection w.r.t a line transformation transformation and symmetry in geometrical shapes

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:

  1. $\dfrac {13}{5}$
  2. $\sqrt {5}$
  3. $5\sqrt {2}$
  4. $5$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The line equation can be rewritten as (2x+3y-5)cos(theta) + (3x-5y+2)sin(theta) = 0. This passes through the intersection of 2x+3y=5 and 3x-5y=-2. Solving this gives P(1, 1). Reflecting P across 4x+6y=23 gives Q, then calculate distance to origin.

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

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

  1. $y = x + \sqrt 3 $
  2. $\sqrt 3 y = x - \sqrt 3 $
  3. $y = \sqrt 3 x - \sqrt 3 $
  4. $\sqrt 3 y = x - 1$
Reveal answer Fill a bubble to check yourself
B Correct answer
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, $\sqrt3 y=x-\sqrt3$

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

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=

  1. $\left( { - 3,\,2\sqrt 2 } \right)$
  2. $\left( {3,\,2\sqrt 2 } \right)$
  3. $\left( {3,\, - 2\sqrt 2 } \right)$
  4. $\left( {3,\,8\sqrt 2 } \right)$
Reveal answer Fill a bubble to check yourself
A Correct answer
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 $(-3, 2\sqrt{2})$.
Multiple choice maths position and movement reflection w.r.t a line transformation transformation and symmetry in geometrical shapes

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

  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. $\left(\dfrac{1}{6}, \dfrac{2}{3}, \dfrac{3}{4}\right)$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The reflection of (x1, y1, z1) in ax+by+cz+d=0 is given by (x-x1)/a = (y-y1)/b = (z-z1)/c = -2(ax1+by1+cz1+d)/(a^2+b^2+c^2). Plugging in the values gives the point (26/7, 15/7, 17/7).