Question

If one end of a focal chord $AB$ of the parabola $y^2=8x$ is at $A\left(\frac{1}{2},-2\right)$, then the equation of a tangent to it at $B$ is

• A. $x+2y+8=0$
• B. $2x-y-24=0$
• C. $x-2y+8=0$
• D. $2x+y-24=0$

Answer 1

The correct option is C

$x-2y+8=0$

Explanation for the correct option.

Step 1: Explanation of the terms required for the given parabola

As the equation of the parabola given here is $y^2=8x$ and the equation of a parabola is given by $y^2=4ax$ then we have

$a=2$, where $a$ is the distance between the focus and the vertex.

Now, in the parametric form, the endpoints of a focal length of a parabola can be written as $A\left(a{t_1}^2,2a{t}_1\right)$ and $B\left(a{t_2}^2,2a{t}_2\right)$.

Also, when $t_1\text{and}{t}_2$ are the endpoints of a focal length of a parabola then the product of the endpoints that is $t_1{t}_2=-1$.

Step 2: Calculation and formation of the equation of a tangent

Since the coordinates of the point $A$ are $\left(\frac{1}{2},-2\right)$ then

$a{t_1}^2=\frac{1}{2}\text{and}2a{t}_1=-2$

From this, we can conclude that $t_1=-\frac{1}{2}$ because $a=2$.

Also, we know that $t_1{t}_2=-1$ then

$t_2=2$

Therefore, the coordinates of a point $B$ are $\left(2{\left(2\right)}^2,2\left(2\right)\left(2\right)\right)$ which is $\left(8,8\right)$.

As the tangent is touching the parabola at point $B$ then its equation will be $t_{2}y=x+a{t_2}^2$

$2y=x+8$ which can be rewritten as $x-2y+8=0$.

Hence, the correct option is (C).