The condition that the line `I x + my = n` be a tangent to the ellipse `(x^2)/(a^2) + (y^2)/(b^2)=1` are .................. .........

Answer

To determine the condition under which the line \( lx + my = n \) is a tangent to the ellipse given by the equation

\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1, \]

we can follow these steps:

### Step 1: Write the equation of the tangent line to the ellipse
The general equation of the tangent to the ellipse at a point can be expressed as:

\[ \frac{xx_1}{a^2} + \frac{yy_1}{b^2} = 1, \]

where \((x_1, y_1)\) is a point on the ellipse.

### Step 2: Rearranging the line equation
The line \( lx + my = n \) can be rearranged into the slope-intercept form:

\[ y = -\frac{l}{m}x + \frac{n}{m}. \]

Here, the slope of the line is \( -\frac{l}{m} \).

### Step 3: Identify the slope of the tangent line
For the line to be tangent to the ellipse, the slope of the line must be equal to the slope of the tangent to the ellipse at the point of tangency. The slope of the tangent line at any point on the ellipse can be derived from the implicit differentiation of the ellipse equation.

### Step 4: Condition for tangency
The condition for the line to be tangent to the ellipse can be derived from the relationship between the coefficients of the line and the ellipse. The condition can be expressed as:

\[ \frac{n^2}{m^2} = \frac{a^2 l^2}{b^2 + a^2 l^2}. \]

This is derived from substituting the values into the ellipse equation and ensuring that the quadratic equation formed has a discriminant equal to zero (which indicates tangency).

### Final Condition
Thus, the condition that the line \( lx + my = n \) is a tangent to the ellipse \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \) is:

\[ \frac{n^2}{m^2} = \frac{a^2 l^2}{b^2 + a^2 l^2}. \]