Coordinate geometry...Find the ratio in which the line joining A(6,5) and B(4,-3) are divided by the line y=2. AB also intersects the x axis at P....

First of all, we have to find out the equation of the line AB, in order to express the condition of intersecting 2 lines, which is: the coordinates of the point M, resulted from the intersection between lines AB and y=2, have to verify the equations of the 2 lines :AB and Y=2.

In order to find out the equation of the line AB:

(xB-xA)/(x-xA) = (yB-yA)/(y-yA)

Now, we'll substitute the values of known coordinates:

(4-6)/(x-6) = (-3-5)/(y-5)

-2/(x-6) = -8/(y-5)

After dividing by (-2):

1/(x-6) = 4/(y-5)

Cross multiplying:

4(x-6)=y-5

y=4x-24+5

y=4x-19, this being the eq. for the line AB.

The M point belongs to AB, only if it's coordinates verifies the equation of the line AB.

yM=4xM-19

Also yM=2.

By substituting yM=2 into yM=4xM-19, the result will be:

2=4xM-19

4xM=21

xM=21/4

M point coordinates are xM=21/4 and yM=2.

Now we have to find out the length of the segment AB and segment AM, in order to decide the ratio the line AB is split by the line y=2.

The AM segment's length:

AM= sqrt[(xM-xA)^2+(yM-yA)^2]

AM= sqrt[(21/4 - 6)^2 + (2-5)^2]

AM= sqrt[(9/16)+9]

AM=3sqrt17/4

AB= sqrt[(xB-xA)^2 + (yB-yA)^2]

AB= sqrt[(4-6)^2 + (-3-5)^2]

AB= sqrt(68)

AB= 2sqrt17

MB=AB-AM=2sqrt17 - 3sqrt17/4

MB=5sqrt17/4

The ratio is : AM/MB=(3sqrt17/4)/(5sqrt17/4)

AM/MB=3/5

To find out the coordinates for P, we have to remember again the condition that P coordinates has to verify the equation for AB line also.

yP=4xP-19

But yP=0, because P is placed on x axis, so:

0=4xP-19

4xP=19

xP=19/4

P coordinates are: xP=19/4 and  yP=0