If the point `M(h, k)` lie on the line `2x+3y=5` such that `|MA-MB|` is maximum where (1, 2) and B(2, 3), then the value of `(h+k)` is

Answer

To solve the problem, we need to find the point \( M(h, k) \) that lies on the line \( 2x + 3y = 5 \) such that the expression \( |MA - MB| \) is maximized, where \( A(1, 2) \) and \( B(2, 3) \).

### Step 1: Write the equation of the line
The line equation is given as:
\[ 2x + 3y = 5 \]

### Step 2: Express \( k \) in terms of \( h \)
From the line equation, we can express \( k \) in terms of \( h \):
\[ 3y = 5 - 2x \implies y = \frac{5 - 2h}{3} \]
Thus, we have:
\[ k = \frac{5 - 2h}{3} \]

### Step 3: Find the distances \( MA \) and \( MB \)
Now, we need to find the distances \( MA \) and \( MB \):
- The distance \( MA \) from \( M(h, k) \) to \( A(1, 2) \) is given by:
\[ MA = \sqrt{(h - 1)^2 + (k - 2)^2} \]
- The distance \( MB \) from \( M(h, k) \) to \( B(2, 3) \) is given by:
\[ MB = \sqrt{(h - 2)^2 + (k - 3)^2} \]

### Step 4: Set up the expression \( |MA - MB| \)
We want to maximize \( |MA - MB| \). To do this, we can consider the squared distances to avoid dealing with the square root:
\[ |MA - MB| = | \sqrt{(h - 1)^2 + (k - 2)^2} - \sqrt{(h - 2)^2 + (k - 3)^2} | \]

### Step 5: Use the collinearity condition
For \( |MA - MB| \) to be maximized, points \( A \), \( B \), and \( M \) should be collinear. The condition for collinearity can be expressed using the determinant:
\[ \begin{vmatrix} 1 & 2 & 1 \\ 2 & 3 & 1 \\ h & k & 1 \end{vmatrix} = 0 \]
Expanding this determinant gives:
\[ 1(3 - k) - 2(2 - h) + 1(2h - 3) = 0 \]
This simplifies to:
\[ 3 - k - 4 + 2h + 2h - 3 = 0 \implies 4h - k - 4 = 0 \implies k = 4h - 4 \]

### Step 6: Solve the system of equations
Now we have two equations:
1. \( k = \frac{5 - 2h}{3} \)
2. \( k = 4h - 4 \)

Setting these equal to each other:
\[ \frac{5 - 2h}{3} = 4h - 4 \]
Multiplying through by 3 to eliminate the fraction:
\[ 5 - 2h = 12h - 12 \]
Rearranging gives:
\[ 5 + 12 = 12h + 2h \implies 17 = 14h \implies h = \frac{17}{14} \]

### Step 7: Find \( k \)
Substituting \( h \) back into one of the equations to find \( k \):
\[ k = 4\left(\frac{17}{14}\right) - 4 = \frac{68}{14} - \frac{56}{14} = \frac{12}{14} = \frac{6}{7} \]

### Step 8: Calculate \( h + k \)
Now, we find \( h + k \):
\[ h + k = \frac{17}{14} + \frac{6}{7} = \frac{17}{14} + \frac{12}{14} = \frac{29}{14} \]

### Final Answer
Thus, the value of \( h + k \) is:
\[ \frac{29}{14} \]