Given points are A\(\left(\frac 12 , \frac 32\right)\) and B(2, - 5)
Let the point P\(\left(\frac 34, \frac 5{12}\right)\) divide AB in ratio m:n.
For point P on the line joined by the points A and B.
\(\frac 34 = \frac {m \times 2 + n \times \frac 12}{m + n}\) .....(1)
And,
\(\frac 5{12} = \frac{m \times (-5) + n \times \frac 32}{m + n}\) ......(2)
Solving 1,
⇒ \(\frac 34 = \frac{\frac{4m + n}2}{m + n}\)
⇒ \(\frac 34 = \frac{{4m + n}}{2(m + n)}\)
⇒ \(\frac 32 = \frac{{4m + n}}{(m + n)}\)
⇒ \(3(m + n) = 8m + 2n\)
\(∴3m + 3n = 8m + 2n\)
\(∴ 5m = n\)
\(∴ \frac mn = \frac 15\)
Hence, ratio is 1:5.
