Question

The coordinates of the point of reflection the origin $(0,0)$in the line $4x - 2y - 5 = 0$
A. $(1,2)$
B. $(2, - 1)$
C. $\left( {\dfrac{4}{5},\dfrac{2}{5}} \right)$
D. $(2,5)$

Answer 1

Hint: Make use of the concept of image of a point with respect to the line mirror
Let the image of $A(x,y)$with respect to the line mirror.
Then it is given by
$\dfrac{{{x_2} - {x_1}}}{a} = \dfrac{{{y_2} - {y_1}}}{b} = \dfrac{{ - 2(a{x_1} + b{y_1} + c)}}{{({a^2} + {b^2})}}$
Also, Let the foot of perpendicular form the point $A({x_1},{y_1})$which is given by
$\dfrac{{{x_3} - {x_1}}}{a} = \dfrac{{{y_3} - {y_1}}}{b} = \dfrac{{(a{x_1} + b{y_1} + c)}}{{{a^2} + {b^2}}}$
So, using the above technique, reflection points will be fixed out.

Complete step by step solution:
Let $Q({x_1},{y_1})$be the reflection of $(0,0)$with respect to the line $4x - 2y - 5 = 0$
$\left[
  a = 4 \\
  b = - 2 \\
  c = - 5 \\
  \right]$ And ${x_2} = 0,{y_2} = 0$

Then, using the formula
$\dfrac{{{x_2} - {x_1}}}{a} = \dfrac{{{y_2} - {y_1}}}{b} = \dfrac{{ - 2(a{x_1} + b{y_1} + c)}}{{({a^2} + {b^2})}}$
We will substitute the value of $a,b,c,{x_2}and{y_2}$,we have
$\dfrac{{{x_1} - 0}}{4} = \dfrac{{{y_1} - 0}}{{ - 2}} = \dfrac{{ - 2\left[ {4(0) + ( - 2)(0) - 5} \right]}}{{{{(4)}^2} + {{( - 2)}^2}}}$
$\dfrac{{{x_1} - 0}}{4} = \dfrac{{{y_1} - 0}}{{ - 2}} = \dfrac{{ - 2\left[ { - 5} \right]}}{{{{(4)}^2} + {{( - 2)}^2}}}$
$ \Rightarrow \dfrac{{{x_1}}}{4} = \dfrac{{{y_1}}}{{ - 2}} = \dfrac{{10}}{{16 + 4}} = \dfrac{{10}}{{20}}$
$ \Rightarrow \dfrac{{{x_1}}}{4} = \dfrac{{ - {y_1}}}{2} = \dfrac{1}{2}$
$ \Rightarrow \dfrac{{{x_1}}}{4} = \dfrac{1}{2},\dfrac{{ - {y_1}}}{2} = \dfrac{1}{2}$
We will do cross multiply the numbers,
$2{x_1} = 4,2{y_1} = - 2$
${x_1} = \dfrac{4}{2},{y_1} = \dfrac{{ - 2}}{2}$
$ \Rightarrow {x_1} = 2$
$ \Rightarrow {y_1} = - 1$
Therefore, the required reflection point is $\left( {2, - 1} \right)$.
Hence, the correct answer is B.


Note: Simply by putting the required value in the formula illustrated in the hint section, we will get the reflection of any point about any line.