Find the equation of the perpendicular to the segment joining the points A(0,4) and B(−5,9) and passing through the point P. Point P divides the segment AB in the ratio 2:3.

First, let's find the coordinates of point P which divides the segment joining A(0,4) and B(−5,9) internally in the ratio m:n=2:3.
Using the section formula:

P(x,y)=(
m+n
mx
2
​
+nx
1
​

​
,
m+n
my
2
​
+ny
1
​

​
)
x=
2+3
2(−5)+3(0)
​
=
5
−10
​
=−2
y=
2+3
2(9)+3(4)
​
=
5
18+12
​
=
5
30
​
=6
So, the coordinates of point P are (−2,6).

Next, find the slope of the line segment AB (m
1
​
):

m
1
​
=
x
2
​
−x
1
​

y
2
​
−y
1
​

​
=
−5−0
9−4
​
=
−5
5
​
=−1
Since the required line is perpendicular to AB, its slope (m
2
​
) satisfies the condition m
1
​
×m
2
​
=−1:

−1×m
2
​
=−1⟹m
2
​
=1
Now, using the point-slope form, the equation of the line passing through P(−2,6) with slope m
2
​
=1 is:

y−y
1
​
=m
2
​
(x−x
1
​
)
y−6=1(x−(−2))
y−6=x+2
x−y=−8