Question

The point R is the reflection of the point (-1,3) in the line 3y+2x=33. Find by calculation the coordinate of R.

Ans : (7,15 )

Answer 1

is it 3y-2x=33?

Answer 2

3y+2x=33

Answer 3

find the distance of the given point (-1,3) to the line 3y+2x=33... that distance is the same distance to the reflection point R

Answer 4

or we can solve the equation of a line containing both R and given point 'cause this line is perpendicular to the given line.... solve the point of intersection and use the distance formula...

Answer 5

Distance \(\bar{D}\) between a point \((x_0,y_0)\) and line \(Ax+By=C\) can be determined using\[\bar{D}=\frac{|Ax_0+By_0+C|}{\sqrt{A^2+B^2}}\]\[\bar{D}=\frac{|2(-1)+3(3)+(-33)|}{\sqrt{(-1)^2+(3)^2}}\]\[\bar{D}=\frac{|-2+9-33|}{\sqrt{1+9}}=\frac{|-26|}{\sqrt{10}}=\frac{26\sqrt{10}}{10}\]\[\bar{D}=\frac{13\sqrt{10}}{5}~units\]

Answer 6

this distance is the same distance we will get to point R on the other side of the given line...

Answer 7

applying the same formula for distance but this time the point we consider is the reflection point R\((x_R,y_R)\)

Answer 8

\[\frac{26}{\sqrt{10}}=\frac{|2x_R+3y_R-33|}{\sqrt{10}}\]\[26=2x_R+3y_R-33\]\[2x_R+3y_R=26+33=59\]\[2x_R+3y_R=59~~~~~(1)\]

Answer 9

this is the first equation only... we need the another equation to be solved in R\((x_R,y_R)\)

Answer 10

let us consider another line where the given point (-1,3) and point R are endpoints, this line is perpendicular to the given line...\[2x+3y=33\]transforming to slope-intercept form \(y=mx+b\)\[3y=-2x+33\]\[y=\left(-\frac{2}{3}\right)x+11\]therefore \(m=-\frac{2}{3}\) and y-intercept b=+11

Answer 11

using point-slope form...the slope of perpendicular lines \(l_1\) and \(l_2\) is given by\[m_1=-\frac{1}{m_2}\]therefore the slope a line we are looking is \[m=-\frac{1}{-\frac{2}{3}}=\frac{3}{2}\]then the line equation will be...\[(y-y_1)=m(x-x_1)\]\[(y-3)=\frac{3}{2}(x-(-1))\]\[2(y-3)=3(x+1)\]\[2y-6=3x+3\]\[3x-2y=-9\]this equation of a line contains the reflection point R... so we can substitute its coordinates in place of x and y....\[3x_R-2y_R=-9~~~~~(2)\]solving simultaneously (1) and (2) will allow you solve the coordinates of point R.

Answer 12

maybe you can continue beyond this point @eunnice ...\(\ddot\smile\)

Answer 13

solve for R\((x_R,y_R)\) using the following equations:\[2x_R+3y_R=59~~~~~(1)\]\[3x_R-2y_R=-9~~~~~(2)\]

Answer 14

line 3x-2y=-9 is the line that joins point (-1,3) and point R... these points are equidistant to given line 3y+2x=33

Answer 15

understand the problem? @eunnice

Answer 16

yes i get it now :) thanks alot!

Answer 17

very good... :-)