Animated Solution for Mathematics - Conic Sections: If one end of a focal chord $AB$ of the parabola $y^2=8x$ is at $A(\frac{1}{2},-2)$, then the equation of the tangent to it at $B$ is :

We are dealing with the parabola defined by the equation:

This is a standard parabola opening to the right, symmetric about the $x$-axis. We are given a point $A(\frac{1}{2},-2)$ on this curve, which serves as one endpoint of a focal chord. Our objective is to determine the equation of the tangent line at the other endpoint of this chord, point $B$.

Every parabola is defined by its focus. By comparing our equation $y^2=8x$ with the standard form $y^2=4ax$, we identify:

The focus $S$ is located at $(a,0)$, which gives us the coordinate $S(2,0)$. This point is the anchor for our focal chord; any line segment connecting two points on the parabola that passes through $(2,0)$ is a focal chord.

We describe any point on the parabola $y^2=4ax$ using the parameter $t$ as $(a{t}^2,2at)$. Given $a=2$, any point on our parabola takes the form $(2t^2,4t)$. For point $A(\frac{1}{2},-2)$, we equate the $y$-coordinates:

For any focal chord of a parabola, the product of the parameters of the endpoints $t_1$ and $t_2$ must satisfy the condition:

Substituting our known value $t_1=-\frac{1}{2}$, we solve for $t_2$:

With the parameter $t_2=2$ identified, we calculate the coordinates of point $B$ using $(2t^2,4t)$:

Thus, point $B$ is $(8,8)$. The equation of the tangent to the parabola $y^2=4ax$ at a point $(x_1,y_1)$ is given by $y{y}_1=2a(x+x_1)$. Substituting $a=2$, $x_1=8$, and $y_1=8$:

Dividing both sides by 4, we obtain $2y=x+8$. Rearranging this into the standard form $Ax+By+C=0$, we reach the final result: