Question

If (-2, 6) is the image of the point (4, 2) with respect to the line L=0, then L=?
a) \[3x-2y+5=0\]
b) \[3x-2y+10=0\]
c) \[2x+3y-5=0\]
d) \[6x-4y-7=0\]

Answer 1

Hint:In the given question, we have been asked to find the equation of the line L. In order to find the equation of the line L, first we need to find the midpoint of the points given in the question, then we have to find the slope and then we form the equation by putting the midpoint and the slope of the line L. We can solve this question by applying the formula of slope-intercept form. The slope- intercept form of a linear equation is: y = mx + b, where m is the slope and b is the y-intercept value.

Complete step by step solution:
Let the points are A (-2, 6) and B (4, 2)
Coordinates of midpoint of a line AB is given by;
\[\Rightarrow mid-po\operatorname{int}=m=\left(
\dfrac{{{x}_{1}}+{{x}_{2}}}{2},\dfrac{{{y}_{1}}+{{y}_{2}}}{2} \right)\]
Where \[\left( {{x}_{1}},{{y}_{1}} \right)\]= (-2, 6) are the coordinates of the point A and \[\left(
{{x}_{2}},{{y}_{2}} \right)\]= (4, 2) are the coordinates of point B.
So, midpoint of AB will lie on line L.
Therefore, midpoint of the line AB is,
\[\Rightarrow m=\left( \dfrac{-2+4}{2},\dfrac{6+2}{2} \right)=\left( \dfrac{2}{2},\dfrac{8}{2}
\right)=\left( 1,4 \right)\]
Therefore, the coordinates of the midpoint of the line AB is (1, 4)
If (-2, 6) is the image of the point (4, 2) with respect to the line L=0
So, L=0 is perpendicular to the line AB and it passes through the midpoint.
Now,
Slope of the line AB =\[\dfrac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}}\], where \[\left( {{x}_{1}},{{y}_{1}}
\right)\]= (-2, 6) are the coordinates of the point A and \[\left( {{x}_{2}},{{y}_{2}} \right)\]= (4, 2) are the coordinates of the point B.
Therefore, slope of AB \[=\dfrac{2-6}{4-\left( -2 \right)}=\dfrac{-4}{4+2}=\dfrac{-4}{6}=\dfrac{-2}{3}\]
Since L is perpendicular to AB.
Therefore, slope of line L = \[\dfrac{3}{2}\]
Thus equation of line L passing through (1, 4) and having the slope \[\dfrac{3}{2}\] is given by,
\[\Rightarrow y-4=\dfrac{3}{2}\left( x-1 \right)\]
\[\Rightarrow 2\left( y-4 \right)=3\left( x-1 \right)\]
\[\Rightarrow 2y-8=3x-3\]
\[\Rightarrow 3x-3-2y+8=0\]
\[\Rightarrow 3x-2y+5=0\]
Hence, the option (a) is the correct answer.

Note: We can solve this question by applying the formula of slope-intercept form. The slope-intercept form of a linear equation is: y = mx + b, where m is the slope and b is the y-intercept value. We should always remember that changing the form of a line’s equation does not change the line. It simply rewrites in a different way.