In what ratio does (0,3) divide the line segment joining A(-2,-1) and B(3,9)?

Question

I learned the formula to find the coordinates that divide a segment with endpoints x1 and x2 with ratio a:b, which is:
(x-coordinate): x1 + (a/(a+b))(x2-x1)
(y-coordinate): same as above but use y-coordinate

If someone knows how to solve find the ratio using this formula, that would also be helpful.

Answer 1

THIS IS WHAT YOU WANT:
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the coordinates of point are (x,y)
x=x1+(a/a+b)(x2-x1)
y=y1+(a/a+b)(y2-y1)

let's supose
p=x-x1 and q=x2-x1
now,

p/q=(x-x1)/(x2-x1)=a/a+b
p/q=a/a+b
aq=p(a+b)
a(q-p)=pb
so b=a(q-p)/p
ratio is
a:a(q-p)/p
1:(q-p)/p
p:(q-p)

since,
q=x2-x1
p=x-x1

the ratio is::
x-x1:(x2-x1-x+x1)
x-x1:x2-x
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SO, THE FORMULA FOR RATIO IN TERMS OF x,x1,x2
x-x1:x2-x

SIMILARLY, THE FORMULA FOR RATIO IN TERMS OF y,y1,y2
y-y1:y2-y

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Answer 2

0 = -2+5a/(a+b) so 5a=2 --> a = 2/5
3 = -1 +10a/(a+b)
4(b+1) = 10
4b+4 = 10
4b = 6
b = 3/2
a/b = (2/5)/(3/2) = 4/15