If `25 p^2+9q^2-r^2-30 p q=0` a point on the line `px+qy+r=0` is

Options

A. `(5,-3)`
B. `(1,2)`
C. `(0,0)`
D. `(5,3)`

Answer

To solve the problem, we need to find a point on the line defined by the equation \( px + qy + r = 0 \) given the condition \( 25p^2 + 9q^2 - r^2 - 30pq = 0 \).

### Step-by-Step Solution:

1. **Rewrite the Given Equation**:
We start with the equation:
\[ 25p^2 + 9q^2 - r^2 - 30pq = 0 \]
Rearranging gives us:
\[ 25p^2 + 9q^2 - 30pq - r^2 = 0 \]

2. **Recognize the Structure**:
Notice that \( 25p^2 - 30pq + 9q^2 \) can be factored. It can be rewritten as:
\[ (5p - 3q)^2 - r^2 = 0 \]
This is a difference of squares.

3. **Factor the Expression**:
The expression can be factored as:
\[ (5p - 3q + r)(5p - 3q - r) = 0 \]
This implies that either:
\[ 5p - 3q + r = 0 \quad \text{or} \quad 5p - 3q - r = 0 \]

4. **Solve for r**:
From the first equation:
\[ r = 3q - 5p \]
From the second equation:
\[ r = 5p - 3q \]

5. **Substitute r into the Line Equation**:
We will substitute \( r \) into the line equation \( px + qy + r = 0 \).

**Case 1**: Using \( r = 3q - 5p \):
\[ px + qy + (3q - 5p) = 0 \]
Rearranging gives:
\[ px + qy - 5p + 3q = 0 \]
Factoring out \( p \) and \( q \):
\[ p(x - 5) + q(y + 3) = 0 \]
Dividing by \( p \):
\[ x - 5 + \frac{q}{p}(y + 3) = 0 \]
This represents a family of lines passing through the fixed point \( (5, -3) \).

**Case 2**: Using \( r = 5p - 3q \):
\[ px + qy + (5p - 3q) = 0 \]
Rearranging gives:
\[ px + qy + 5p - 3q = 0 \]
Factoring out \( p \) and \( q \):
\[ p(x + 5) + q(y - 3) = 0 \]
Dividing by \( p \):
\[ x + 5 + \frac{q}{p}(y - 3) = 0 \]
This represents a family of lines passing through the fixed point \( (-5, 3) \).

6. **Identify Points**:
The points we found are:
- From Case 1: \( (5, -3) \)
- From Case 2: \( (-5, 3) \)

7. **Check Options**:
The options provided were:
1. \( (5, -3) \)
2. \( (1, 2) \)
3. \( (0, 0) \)
4. \( (5, 3) \)

The point \( (5, -3) \) is present in the options.

### Final Answer:
The point on the line \( px + qy + r = 0 \) is \( (5, -3) \).