To find the condition that the straight line `(x)/(f) + (y)/(g) = 1 ` may be a tangent

Answer

To find the condition that the straight line \(\frac{x}{f} + \frac{y}{g} = 1\) may be a tangent to the parabola \(y^2 = 4ax\), we can follow these steps:

### Step-by-Step Solution:

1. **Rewrite the Line Equation**:
Start with the line equation:
\[ \frac{x}{f} + \frac{y}{g} = 1 \]
Rearranging gives:
\[ y = -\frac{g}{f}x + g \]
This is in the slope-intercept form \(y = mx + c\) where \(m = -\frac{g}{f}\) and \(c = g\).

2. **General Equation of Tangent to the Parabola**:
The general equation of the tangent to the parabola \(y^2 = 4ax\) is given by:
\[ y = mx + \frac{a}{m} \]
where \(m\) is the slope of the tangent.

3. **Set the Slopes Equal**:
For the line to be a tangent to the parabola, the slopes must be equal. Thus, we equate the slopes:
\[ -\frac{g}{f} = m \]

4. **Substituting for \(m\)**:
Substitute \(m\) into the tangent equation:
\[ y = -\frac{g}{f}x + \frac{a}{-\frac{g}{f}} = -\frac{g}{f}x - \frac{af}{g} \]

5. **Comparing the y-intercepts**:
We can also compare the y-intercepts from both equations. From the line, the y-intercept is \(g\), and from the tangent, it is \(-\frac{af}{g}\). Setting these equal gives:
\[ g = -\frac{af}{g} \]

6. **Cross Multiply**:
Cross-multiplying gives:
\[ g^2 = -af \]

7. **Rearranging**:
Rearranging this equation leads to:
\[ g^2 + af = 0 \]

### Final Condition:
The condition for the line \(\frac{x}{f} + \frac{y}{g} = 1\) to be a tangent to the parabola \(y^2 = 4ax\) is:
\[ g^2 + af = 0 \]