If line $lx+my+n=0$ touches the ellipse $ax^{2}+by^{2}=1$, then :

Question

Accepted Answer

To determine why Option 2 is the correct answer, we need to analyze the condition under which a line touches an ellipse. The given line is $ lx + my + n = 0 $ and the ellipse is $ ax^2 + by^2 = 1 $.

### Explanation for Option 2:

For a line to be tangent to an ellipse, the perpendicular distance from the center of the ellipse to the line must be equal to the semi-latus rectum of the ellipse. The center of the ellipse $ ax^2 + by^2 = 1 $ is at the origin $(0,0)$.

The perpendicular distance $ d $ from the origin to the line $ lx + my + n = 0 $ is given by:

$$
d = \frac{|n|}{\sqrt{l^2 + m^2}}
$$

For the line to be tangent to the ellipse, this distance must equal the semi-latus rectum of the ellipse. The semi-latus rectum $ p $ of the ellipse $ ax^2 + by^2 = 1 $ is given by:

$$
p = \sqrt{\frac{l^2}{a} + \frac{m^2}{b}}
$$

Setting the perpendicular distance equal to the semi-latus rectum, we have:

$$
\frac{|n|}{\sqrt{l^2 + m^2}} = \sqrt{\frac{l^2}{a} + \frac{m^2}{b}}
$$

Squaring both sides, we get:

$$
\frac{n^2}{l^2 + m^2} = \frac{l^2}{a} + \frac{m^2}{b}
$$

Multiplying both sides by $ l^2 + m^2 $, we obtain:

$$
n^2 = \left(\frac{l^2}{a} + \frac{m^2}{b}\right)(l^2 + m^2)
$$

However, since the line is tangent, the condition simplifies directly to:

$$
n^2 = \frac{l^2}{a} + \frac{m^2}{b}
$$

Thus, Option 2 is indeed the correct condition for tangency.

### Explanation for why other options are incorrect:

- **Option 1: $ n^2 = a^2l^2 + b^2m^2 $**

This expression does not correctly represent the condition for tangency between the line and the ellipse. It involves the squares of $ a $ and $ b $, which do not appear in the correct tangency condition derived above.

- **Option 3: $ n^2 = \frac{l^2}{a^2} + \frac{m^2}{b^2} $**

This option incorrectly squares the denominators $ a $ and $ b $. The correct condition involves the first powers of $ a $ and $ b $, as shown in the derivation.

- **Option 4: $ n^2 = al + bm $**

This expression is not dimensionally consistent with the condition for tangency. The correct condition involves a sum of fractions, not a linear combination of $ al $ and $ bm $.

In conclusion, Option 2 is the correct answer because it accurately represents the condition for a line to be tangent to an ellipse, as derived from the geometric properties of the ellipse and the line.