Question

A diagram shows lines x-2y-4=0, x+ y=5 and the point P (1,1). The line is drawn from P to intersect with x-2y-4= 0 at Q, and with x+y=5 at point R, so the P is midpoint of the QR. Find the coordinates for the point R and Q.

Answer 1

a = x coordinate of point Q on line x - 2 y - 4 = 0

b = y coordinate of point Q on line x - 2 y - 4 = 0

Put this coordinates in equation:

x - 2 y - 4 = 0

a - 2 b - 4 = 0

Add 4 to both sides

a - 2 b = 4

If P is the midpoint of QR, then:

( R + Q ) / 2 = P

Multiply both sides by 2

R + Q = 2 P

Subtract Q to both sides

R = 2 P - Q

x coordinate of point P = 1

y coordinate of point P = 1

x coordinate of point Q = a

y coordinate of point Q = b

R = 2 P - Q

R = 2 ( 1 , 1 ) - ( a , b ) = ( 2 , 2 ) - ( a , b ) = ( 2 - a ) , ( 2 - b )

You know this point also satisfies the equation:

x + y = 5

Put x coordinate of point x = 2 - a and y coordinate of point y = 2 - b in this equation:

( 2 - a ) + ( 2 - b ) = 5

2 - a + 2 - b = 5

4 - a - b = 5

Subtract 4 to both sides

- a - b = 1

Multiply both sides by - 1

a + b = - 1

To find the solution you can subtract the first from the second:

( a + b = - 1 ) - ( a - 2 b = 4 )

a + b - ( a - 2 b ) = - 1 - 4

a + b - a + 2 b = - 5

3 b = - 5

b = - 5 / 3

Now:

a + b = - 1

Subtract b a to both sides

a = - 1 - b

a = - 1 - ( - 5 / 3 ) = - 3 / 3 + 5 / 3 = 2 / 3

The point Q is ( a , b ) = ( 2 / 3, - 5 / 3 )

The point R is:

R = ( 2 - a , 2 - b ) = ( 2 - 2 / 3, ) , [ 2 - ( - 5 / 3 ) ] =

( 6 / 3 - 2 / 3 , 2 + 5 / 3 ) = ( 4 / 3 , 6 / 3 + 5 / 3 ) = ( 4 / 3 , 11 / 3 )

The coordinates of Q and R are:

Q ( 2 / 3, - 5 / 3 )

R ( 4 / 3, 11 / 3 )