Animated Solution for Mathematics - Conic Sections: The equation of a tangent to the hyperbola $4x^2-5y^2=20$ parallel to the line $x-y=2$ is :

To begin, we must transform the given hyperbola equation into its standard form. The equation $4x^2-5y^2=20$ is currently in a raw state.

Dividing the entire equation by $20$, we obtain:

This simplifies to the standard form:

By comparing this to the general form $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$, we identify our parameters as $a^2=5$ and $b^2=4$.

Next, we determine the slope of the reference line $x-y=2$. Rearranging this into the slope-intercept form $y=mx+c$, we get $y=x-2$.

The coefficient of $x$ reveals that the slope is $m=1$. Since our required tangent must be parallel to this line, it must share the same slope, $m=1$.

For any hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$, the condition for a line $y=mx+c$ to be a tangent is defined by the formula:

Substituting our known values $m=1$, $a^2=5$, and $b^2=4$ into this expression, we calculate:

We now substitute the values of $m$ and $c$ back into the general line equation $y=mx+c$. This yields two possible tangent lines:

Rewriting these in the standard form $Ax+By+C=0$, we obtain:

The resulting equations represent the two tangents to the hyperbola parallel to the given line. Thus, the required tangent is $x-y+1=0$ (or $x-y-1=0$ depending on the specific options provided).