The image of the origin with reference to the line 4x+3y -25=0 is

Options

A. `(-8,6)`
B. 8,6
C. `-3,4`
D. `8,-6`

Explanation

To find the image of the origin (0, 0) with respect to the line given by the equation \(4x + 3y - 25 = 0\), we can follow these steps:

### Step 1: Identify the equation of the line
The equation of the line is given as:
\[ 4x + 3y - 25 = 0 \]

### Step 2: Set the coordinates of the image
Let the coordinates of the image of the origin be \((h, k)\).

### Step 3: Find the midpoint of the origin and its image
The midpoint \(M\) of the segment joining the origin \((0, 0)\) and the image \((h, k)\) is given by:
\[ M = \left(\frac{h + 0}{2}, \frac{k + 0}{2}\right) = \left(\frac{h}{2}, \frac{k}{2}\right) \]

### Step 4: Substitute the midpoint into the line equation
Since the midpoint lies on the line, we substitute \(\left(\frac{h}{2}, \frac{k}{2}\right)\) into the line equation:
\[ 4\left(\frac{h}{2}\right) + 3\left(\frac{k}{2}\right) - 25 = 0 \]
This simplifies to:
\[ 2h + \frac{3k}{2} - 25 = 0 \]
Multiplying through by 2 to eliminate the fraction gives:
\[ 4h + 3k - 50 = 0 \quad \text{(Equation 1)} \]

### Step 5: Find the slope of the line
The slope of the line \(4x + 3y - 25 = 0\) can be calculated from the coefficients:
\[ \text{slope} = -\frac{\text{coefficient of } x}{\text{coefficient of } y} = -\frac{4}{3} \]

### Step 6: Find the slope of the line joining the origin and the image
The slope of the line joining the origin \((0, 0)\) and the image \((h, k)\) is:
\[ \text{slope} = \frac{k - 0}{h - 0} = \frac{k}{h} \]

### Step 7: Set up the perpendicularity condition
Since the two lines are perpendicular, the product of their slopes must equal \(-1\):
\[ \frac{k}{h} \cdot \left(-\frac{4}{3}\right) = -1 \]
This simplifies to:
\[ \frac{4k}{3h} = 1 \implies 4k = 3h \quad \text{(Equation 2)} \]

### Step 8: Solve the equations
From Equation 2, we can express \(k\) in terms of \(h\):
\[ k = \frac{3h}{4} \]

Substituting this into Equation 1:
\[ 4h + 3\left(\frac{3h}{4}\right) - 50 = 0 \]
This simplifies to:
\[ 4h + \frac{9h}{4} - 50 = 0 \]
Multiplying through by 4 to eliminate the fraction:
\[ 16h + 9h - 200 = 0 \implies 25h = 200 \implies h = 8 \]

### Step 9: Find \(k\)
Now substituting \(h = 8\) back into the equation for \(k\):
\[ k = \frac{3 \cdot 8}{4} = 6 \]

### Step 10: Conclusion
The coordinates of the image of the origin with respect to the line \(4x + 3y - 25 = 0\) are:
\[ (h, k) = (8, 6) \]