Question

The Point A(2,-3) is reflected in the x axis onto the point A'.Then the point A' is reflected in the line x=4 onto the point A''.

1)Write the coordinates of A' and A''

2)Find the ratio in which the line segment AA'' is divided by the x-axis.Also Find the coordinates of the point of division.

Answer 1

a) given that A(2,-3).
* To find the reflection of a point on x axis change the sign of y coordinate and retain the x coordinate.
Then A'=(2,3)
* To find the reflection of a point P(x,y) on a line in the form x=a, P"=(-x+2a,y)
Here A"=(-2+2*4,3)
A"=(6,3)

b)let the line segment AA" is divided by the x axis in the ratio m:1.
Then the point of division P is given by
P(mx2+x1/m+1,my2+y1/m+1)
Here A(2,-3) and A"(6,3)
x1=2 y1=-3
x2=6 y2=3
P(m*6+2/m+1,m*3+-3/m+1)
Since the point P is in x axis y coordinate is zero.
ie., m*3+-3/m+1=0
3m-3/m+1=0
3m-3=0
m=3/3=1
The ratio at which the x axis cuts AA" is 1:1.
Point of division P(1*6+2/1+1,0)
P(4,0)