If the line `x/a+y/b=1` passes through the points `(2,-3)` and `(4,-5)` then (a,b) is (i) `(1,1)` (ii) `(-1,1)` (iii) `(1,-1)` (iv) `(-1,-1)`

Options

A. `(1,1)`
B. `(-1,1)`
C. `(1,-1)`
D. `(-1,-1)`

Answer

To solve the problem, we need to determine the values of \(a\) and \(b\) such that the line given by the equation

\[ \frac{x}{a} + \frac{y}{b} = 1 \]

passes through the points \((2, -3)\) and \((4, -5)\).

### Step 1: Substitute the first point into the line equation

Substituting the point \((2, -3)\) into the line equation:

\[ \frac{2}{a} + \frac{-3}{b} = 1 \]

This can be rearranged to form the first equation:

\[ \frac{2}{a} - \frac{3}{b} = 1 \quad \text{(Equation 1)} \]

### Step 2: Substitute the second point into the line equation

Now, substitute the point \((4, -5)\) into the line equation:

\[ \frac{4}{a} + \frac{-5}{b} = 1 \]

This can be rearranged to form the second equation:

\[ \frac{4}{a} - \frac{5}{b} = 1 \quad \text{(Equation 2)} \]

### Step 3: Solve the system of equations

Now we have a system of two equations:

1. \(\frac{2}{a} - \frac{3}{b} = 1\) (Equation 1)
2. \(\frac{4}{a} - \frac{5}{b} = 1\) (Equation 2)

To eliminate one variable, we can multiply Equation 1 by 2:

\[ \frac{4}{a} - \frac{6}{b} = 2 \quad \text{(Equation 3)} \]

### Step 4: Subtract Equation 2 from Equation 3

Now, subtract Equation 2 from Equation 3:

\[ \left(\frac{4}{a} - \frac{6}{b}\right) - \left(\frac{4}{a} - \frac{5}{b}\right) = 2 - 1 \]

This simplifies to:

\[ -\frac{6}{b} + \frac{5}{b} = 1 \]

Which further simplifies to:

\[ -\frac{1}{b} = 1 \]

### Step 5: Solve for \(b\)

From the equation \(-\frac{1}{b} = 1\), we can solve for \(b\):

\[ b = -1 \]

### Step 6: Substitute \(b\) back into Equation 1 to find \(a\)

Now, substitute \(b = -1\) back into Equation 1:

\[ \frac{2}{a} - \frac{3}{-1} = 1 \]

This simplifies to:

\[ \frac{2}{a} + 3 = 1 \]

Rearranging gives:

\[ \frac{2}{a} = 1 - 3 \]

\[ \frac{2}{a} = -2 \]

### Step 7: Solve for \(a\)

Now, solving for \(a\):

\[ a = -1 \]

### Conclusion

Thus, we have found:

\[ (a, b) = (-1, -1) \]

The correct option is (iv) \((-1, -1)\).

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