The orthocentre of the triangle formed by the lines xy = 0 and `2x+3y-5=0` is

Options

A. (2, 3)
B. (3, 2)
C. (0, 0)
D. (5, -5)

Answer

To find the orthocenter of the triangle formed by the lines \( xy = 0 \) and \( 2x + 3y - 5 = 0 \), we can follow these steps:

### Step 1: Identify the lines
The equation \( xy = 0 \) represents two lines:
1. \( x = 0 \) (the y-axis)
2. \( y = 0 \) (the x-axis)

The second line is given by \( 2x + 3y - 5 = 0 \).

### Step 2: Find the intersection points
Next, we need to find the points where these lines intersect to form the vertices of the triangle.

1. **Intersection of \( x = 0 \) and \( 2x + 3y - 5 = 0 \)**:
- Substitute \( x = 0 \) into the second equation:
\[ 2(0) + 3y - 5 = 0 \implies 3y = 5 \implies y = \frac{5}{3} \]
- So, the intersection point is \( (0, \frac{5}{3}) \).

2. **Intersection of \( y = 0 \) and \( 2x + 3y - 5 = 0 \)**:
- Substitute \( y = 0 \) into the second equation:
\[ 2x + 3(0) - 5 = 0 \implies 2x = 5 \implies x = \frac{5}{2} \]
- So, the intersection point is \( (\frac{5}{2}, 0) \).

3. **Intersection of \( x = 0 \) and \( y = 0 \)**:
- The intersection point is \( (0, 0) \).

### Step 3: Identify the vertices of the triangle
The vertices of the triangle formed by the lines are:
1. \( A(0, 0) \)
2. \( B(0, \frac{5}{3}) \)
3. \( C(\frac{5}{2}, 0) \)

### Step 4: Find the orthocenter
The orthocenter of a triangle is the point where the altitudes intersect. In this case, since one vertex is at the origin \( (0, 0) \), we can observe the following:

- The altitude from vertex \( A(0, 0) \) to side \( BC \) is a vertical line along the y-axis.
- The altitude from vertex \( B(0, \frac{5}{3}) \) to side \( AC \) is a horizontal line along the x-axis.
- The altitude from vertex \( C(\frac{5}{2}, 0) \) will be a line perpendicular to \( AB \).

Since \( A(0, 0) \) is the vertex where the right angle is formed (as the lines \( x = 0 \) and \( y = 0 \) are perpendicular), the orthocenter of the triangle is at the vertex \( A \).

Thus, the orthocenter of the triangle is:
\[ \text{Orthocenter} = (0, 0) \]

### Final Answer
The orthocenter of the triangle formed by the lines \( xy = 0 \) and \( 2x + 3y - 5 = 0 \) is \( (0, 0) \).

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