Animated Solution for Mathematics - Conic Sections: If $x=9$ is the chord of contact of the hyperbola $x^2-y^2=9$, then the equation of the corresponding pair of tangents is

The hyperbola is defined by the equation $x^2-y^2=9$. We are given a chord of contact, represented by the line $x=9$, which connects the two points where tangents from an external point touch the hyperbola.

Our objective is to determine the combined equation of these two tangents.

To identify the points of contact, we find the intersection of the chord $x=9$ and the hyperbola $x^2-y^2=9$. Substituting $x=9$ into the hyperbola equation yields:

Solving for $y$, we find $y=\pm 6\sqrt{2}$. Thus, the two points of contact are $P(9,6\sqrt{2})$ and $Q(9,-6\sqrt{2})$.

For a hyperbola of the form $x^2-y^2=a^2$, the equation of the tangent at a point $(x_1,y_1)$ is given by $x{x}_1-y{y}_1=a^2$. Here, $a^2=9$.

For point $P(9,6\sqrt{2})$, the tangent equation is:

Dividing by $3$, we obtain the simplified form:

For point $Q(9,-6\sqrt{2})$, the tangent equation is:

Dividing by $3$, we obtain:

To find the combined equation of the two tangents, we multiply the individual equations:

We can group the terms to utilize the difference of squares identity, $(a-b)(a+b)=a^2-b^2$, where $a=(3x-3)$ and $b=2\sqrt{2}y$:

Expanding the terms, we get:

Rearranging into the final standard form, the combined equation of the tangents is: