The coordinates of the orthocentre of the triangle whose sides lie along the lines `x=0,y=0` and `3x-5y+7=0` is (i) `(0,(7)/(5))` (ii) `(-(7)/(3),0)` (iii) `(0,0)` (iv) `(-(7)/(5),(7)/(5))`

Options

A. `(0,(7)/(5))`
B. `(-(7)/(3),0)`
C. `(0,0)`
D. `(-(7)/(5),(7)/(5))`

Answer

To find the coordinates of the orthocenter of the triangle formed by the lines \( x = 0 \), \( y = 0 \), and \( 3x - 5y + 7 = 0 \), we can follow these steps:

### Step 1: Identify the lines
The lines given are:
1. \( x = 0 \) (the y-axis)
2. \( y = 0 \) (the x-axis)
3. \( 3x - 5y + 7 = 0 \) (a line in slope-intercept form)

### Step 2: Find the intersection points
The orthocenter of a triangle formed by two perpendicular lines and a third line is the intersection of the two perpendicular lines.

- The intersection of \( x = 0 \) and \( y = 0 \) is:
\[ (0, 0) \]

### Step 3: Check if the third line intersects the axes
Next, we need to find the points where the line \( 3x - 5y + 7 = 0 \) intersects the axes.

1. **Finding the y-intercept** (set \( x = 0 \)):
\[ 3(0) - 5y + 7 = 0 \implies -5y + 7 = 0 \implies 5y = 7 \implies y = \frac{7}{5} \]
So, the y-intercept is \( (0, \frac{7}{5}) \).

2. **Finding the x-intercept** (set \( y = 0 \)):
\[ 3x - 5(0) + 7 = 0 \implies 3x + 7 = 0 \implies 3x = -7 \implies x = -\frac{7}{3} \]
So, the x-intercept is \( (-\frac{7}{3}, 0) \).

### Step 4: Determine the orthocenter
Since the lines \( x = 0 \) and \( y = 0 \) are perpendicular, the orthocenter of the triangle formed by these lines and the line \( 3x - 5y + 7 = 0 \) is the intersection of the two perpendicular lines, which is:
\[ (0, 0) \]

### Conclusion
Thus, the coordinates of the orthocenter of the triangle are:
\[ \boxed{(0, 0)} \]