Question

$A(3,2)$ and $B$ are two points in the $xy$-plane. If the points $C\left(\frac{9}{2},\alpha\right)$ and $D(\beta,8)$ divide $AB$ in the ratio $3:1$ internally and externally respectively, then the coordinates of point $B$ are:

• A. $(5,6)$
• B. $(5,-6)$
• C. $(-5,-6)$
• D. $(-5,6)$

Hint:

When one coordinate of the dividing point is given, use only that coordinate in the section formula. Here, the x-coordinate of the internally dividing point immediately gives \[ \frac{9}{2} = \frac{3x+3}{4} \] which yields $x=5$. Similarly, the y-coordinate of the externally dividing point gives $y=6$. This approach avoids unnecessary calculations involving $\alpha$ and $\beta$.

Answer 1

The Correct Option is A

Concept: The coordinates of a point dividing a line segment joining two points can be determined using the Section Formula. For two points \[ A(x_1,y_1) \quad \text{and} \quad B(x_2,y_2), \] if a point divides the line segment internally in the ratio $m:n$, then its coordinates are \[ \left( \frac{mx_2+nx_1}{m+n}, \frac{my_2+ny_1}{m+n} \right). \] If a point divides the line segment externally in the ratio $m:n$, then its coordinates are \[ \left( \frac{mx_2-nx_1}{m-n}, \frac{my_2-ny_1}{m-n} \right). \] In this problem, we use both the internal and external section formula to determine the unknown coordinates of point $B$.

Step 1: Assume the coordinates of point $B$. Let \[ B=(x,y). \] The coordinates of point $A$ are given as \[ A=(3,2). \]

Step 2: Use the internal division condition involving point $C$. The point \[ C\left(\frac{9}{2},\alpha\right) \] divides the line segment $AB$ internally in the ratio $3:1$. Applying the internal section formula to the x-coordinate: \[ \frac{9}{2} = \frac{3x+1(3)}{3+1}. \] Thus, \[ \frac{9}{2} = \frac{3x+3}{4}. \] Cross-multiplying, \[ 18=3x+3. \] Subtracting $3$ from both sides, \[ 15=3x. \] Therefore, \[ x=5. \] Hence, the x-coordinate of point $B$ is \[ 5. \]

Step 3: Use the external division condition involving point $D$. The point \[ D(\beta,8) \] divides the line segment $AB$ externally in the ratio $3:1$. Applying the external section formula to the y-coordinate: \[ 8 = \frac{3y-1(2)}{3-1}. \] Therefore, \[ 8 = \frac{3y-2}{2}. \] Cross-multiplying, \[ 16=3y-2. \] Adding $2$ to both sides, \[ 18=3y. \] Hence, \[ y=6. \] Thus, the y-coordinate of point $B$ is \[ 6. \]

Step 4: Write the coordinates of point $B$. From Steps 2 and 3, \[ x=5, \qquad y=6. \] Therefore, \[ B=(5,6). \] Hence, the required coordinates of point $B$ are \[ \boxed{(5,6)}. \]