The line $\frac{l}{r}=A~cos~ heta+B~sin~ heta$ will touch the conic $\frac{l}{r}=1+e~cos~ heta$ if:

Q: The line $\frac{l}{r}=A~cos~\theta+B~sin~\theta$ will touch the conic $\frac{l}{r}=1+e~cos~\theta$ if:

The line $\frac{l}{r}=A~cos~\theta+B~sin~\theta$ will touch the conic $\frac{l}{r}=1+e~cos~\theta$ if:

Question

The line $\frac{l}{r}=A~cos~	heta+B~sin~	heta$ will touch the conic $\frac{l}{r}=1+e~cos~	heta$ if:

Accepted Answer

To determine why Option 2 is the correct answer, we need to analyze the conditions under which the given line will touch the conic. The line is given by the equation:

$$
\frac{l}{r} = A \cos \theta + B \sin \theta
$$

The conic is given by the equation:

$$
\frac{l}{r} = 1 + e \cos \theta
$$

For the line to touch the conic, it must be tangent to the conic, meaning they intersect at exactly one point. This condition is satisfied when the discriminant of the resulting quadratic equation (formed by equating the two expressions for $\frac{l}{r}$) is zero.

Equating the two expressions, we have:

$$
A \cos \theta + B \sin \theta = 1 + e \cos \theta
$$

Rearranging terms gives:

$$
(A - e) \cos \theta + B \sin \theta = 1
$$

This is a linear equation in terms of $\cos \theta$ and $\sin \theta$. For this line to be tangent to the conic, the coefficients must satisfy the condition for tangency of a line to a conic. The general condition for tangency of a line $l = A \cos \theta + B \sin \theta = C$ to a conic $r = 1 + e \cos \theta$ is:

$$
(A - e)^2 + B^2 = C^2
$$

In our case, $C = 1$, so the condition becomes:

$$
(A - e)^2 + B^2 = 1
$$

This matches exactly with Option 2, confirming that it is the correct answer.

Now, let's examine why the other options are incorrect:

1. **Option 1: $(A+e)^2 + B^2 = 1$**

This condition would imply that the line is tangent to a different conic, specifically one where the eccentricity term $e$ is added to $A$ rather than subtracted. This does not match the condition derived from the given conic equation, hence it is incorrect.

2. **Option 3: $A + e + B = 0$**

This is a linear condition rather than a quadratic one, and it does not relate to the tangency condition derived from the conic equation. It does not ensure that the line is tangent to the conic, so it is incorrect.

3. **Option 4: $A - e - B = 0$**

Similar to Option 3, this is also a linear condition. It does not satisfy the quadratic tangency condition required for the line to touch the conic. Therefore, it is incorrect.

In conclusion, Option 2 is the correct answer because it satisfies the necessary condition for the line to be tangent to the given conic, ensuring that they intersect at exactly one point.