Question

If the point \(P(a,b)\) divides the line joining the points \(A(-3,4)\), \(B(5,2)\) internally in the ratio \(2:3\), then \(b-a=\)

• A. 4
• B. 3
• C. 2
• D. 1

Hint:

For internal division in the ratio \(m:n\), \[ \boxed{ \left( \frac{mx_2+nx_1}{m+n}, \frac{my_2+ny_1}{m+n} \right) } \] Remember: multiply each endpoint by the opposite ratio.

Answer 1

The Correct Option is B

Concept: If a point \(P(x,y)\) divides the line joining \(A(x_1,y_1)\) and \(B(x_2,y_2)\) internally in the ratio \(m:n\), then by the section formula, \[ \boxed{ P\left( \frac{mx_2+nx_1}{m+n}, \frac{my_2+ny_1}{m+n} \right) } \]

Step 1: Substitute the given values.
Here, \[ A(-3,4),\qquad B(5,2), \] and the ratio is \[ 2:3. \] Hence, \[ a=\frac{2(5)+3(-3)}{2+3} =\frac{10-9}{5} =\frac15, \] and \[ b=\frac{2(2)+3(4)}{5} =\frac{4+12}{5} =\frac{16}{5}. \]

Step 2: Find \(b-a\).
\[ b-a =\frac{16}{5}-\frac15 =\frac{15}{5} =3. \]

Step 3: Final conclusion.
Therefore, \[ \boxed{b-a=3.} \]