To find the specific equation of the curve in the form ax² + by² + f = 0 that passes through the points (4, 0) and (0, 3), we can start by substituting these points into the general equation to derive the values of a and b.
Let’s assume the curve’s equation is given by:
ax² + by² + f = 0
1. **Substituting point (4, 0)**:
Substituting x = 4 and y = 0 gives us:
a(4)² + b(0)² + f = 0
16a + f = 0 (1)
2. **Substituting point (0, 3)**:
Now using x = 0 and y = 3:
a(0)² + b(3)² + f = 0
9b + f = 0 (2)
3. **Solving the equations**:
From equation (1): f = -16a
Substituting f in equation (2):
9b – 16a = 0
Rearranging gives us:
b = rac{16a}{9}
4. **Choosing a value for a**:
To find a specific equation, we will assign a convenient value to a and find b. Let’s take a = 9:
b = rac{16(9)}{9} = 16
5. **Resulting equation**:
Substituting a and b back into the equation:
9x² + 16y² – 144 = 0
Thus, the curve can be expressed as:
9x² + 16y² = 144
6. **Conclusion**:
Therefore, the specific equation of the curve that passes through the points (4, 0) and (0, 3) is Option a: 9x² + 16y² = 144.