We are given an ellipse with equation $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$, where $a \neq b$. We need to find the equation of the set of all points $(x, y)$ from which there are two tangents to the curve such that the slopes of the tangents are (i) reciprocals and (ii) negative reciprocals.

1. Problem Description

We are given an ellipse with equation $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$, where $a \neq b$. We need to find the equation of the set of all points $(x, y)$ from which there are two tangents to the curve such that the slopes of the tangents are (i) reciprocals and (ii) negative reciprocals.

2. Solution Steps

Let the equation of a tangent to the ellipse be $y = mx + c$.

For the line to be a tangent, we must have $c^2 = a^2m^2 + b^2$.

So the equation of the tangent becomes $y = mx \pm \sqrt{a^2m^2 + b^2}$.

Let $(x_1, y_1)$ be a point from which two tangents can be drawn. Then the equation

$y_1 = mx_1 \pm \sqrt{a^2m^2 + b^2}$ must have two real solutions for $m$.

Rearranging, we get $y_1 - mx_1 = \pm \sqrt{a^2m^2 + b^2}$.

Squaring both sides, we have $(y_1 - mx_1)^2 = a^2m^2 + b^2$, which simplifies to

$y_1^2 - 2mx_1y_1 + m^2x_1^2 = a^2m^2 + b^2$.

This gives us the quadratic equation in $m$:

$m^2(x_1^2 - a^2) - 2mx_1y_1 + (y_1^2 - b^2) = 0$.

Let $m_1$ and $m_2$ be the two roots of this equation, which are the slopes of the two tangents.

(i) If the slopes are reciprocals, then $m_1m_2 = 1$.

We know that the product of the roots of the quadratic $Ax^2 + Bx + C = 0$ is $C/A$.

So, $m_1m_2 = \frac{y_1^2 - b^2}{x_1^2 - a^2} = 1$.

Therefore, $y_1^2 - b^2 = x_1^2 - a^2$, which implies $x_1^2 - y_1^2 = a^2 - b^2$.

Thus, the equation of the set of points is $x^2 - y^2 = a^2 - b^2$.

(ii) If the slopes are negative reciprocals, then $m_1m_2 = -1$.

So, $\frac{y_1^2 - b^2}{x_1^2 - a^2} = -1$.

Therefore, $y_1^2 - b^2 = -x_1^2 + a^2$, which implies $x_1^2 + y_1^2 = a^2 + b^2$.

Thus, the equation of the set of points is $x^2 + y^2 = a^2 + b^2$.

3. Final Answer

(i) If the slopes are reciprocals, the equation is $x^2 - y^2 = a^2 - b^2$.

(ii) If the slopes are negative reciprocals, the equation is $x^2 + y^2 = a^2 + b^2$.