Find the ratio in which the point \(P\big(\dfrac{3}{4},\dfrac{5}{12}\big)\) divides the segment joining \(A\big(\dfrac{1}{2},\dfrac{3}{2}\big)\) and \(B(2,-5)\).
\(1:5\) (\(AP:PB\)).
Step 1: Recall the section formula.
If a point \(P(x,y)\) divides a line joining two points \(A(x_1,y_1)\) and \(B(x_2,y_2)\) in the ratio \(m:n\), then
\[ x = \dfrac{mx_2 + nx_1}{m+n}, \quad y = \dfrac{my_2 + ny_1}{m+n} \]
Step 2: Write down the given values.
Step 3: Apply the section formula for the \(x\)-coordinate.
\[ \tfrac{3}{4} = \dfrac{m(2) + n(\tfrac{1}{2})}{m+n} \]
Step 4: Simplify the numerator.
\[ \tfrac{3}{4} = \dfrac{2m + \tfrac{n}{2}}{m+n} \]
Step 5: Remove the fraction in the numerator.
Multiply numerator and denominator by 2: \[ \tfrac{3}{4} = \dfrac{4m + n}{2m + 2n} \]
Step 6: Cross multiply.
\[ 3(2m + 2n) = 4(4m + n) \]
\[ 6m + 6n = 16m + 4n \]
Step 7: Rearrange terms.
\[ 6n - 4n = 16m - 6m \] \[ 2n = 10m \]
Step 8: Simplify the ratio.
\[ n = 5m \quad \Rightarrow \quad m:n = 1:5 \]
Step 9: Verify with the \(y\)-coordinate (optional check).
Using \(y = \tfrac{my_2 + ny_1}{m+n}\): Substitute \(m:n = 1:5\), You will get \(y = \tfrac{5}{12}\), which matches the given \(P\).
Final Answer: The point \(P\) divides \(AB\) in the ratio \(1:5\) (that is, \(AP:PB = 1:5\)).