Circles

To solve the problem, we need to determine the value of such that the line (or ) is the directrix of the given parabola .

A parabola with horizontal axis can be represented by the general equation:

Here, the vertex is at , the focus is at , and the directrix is the line .

Comparing this to the given equation of the parabola, , we can rewrite this as:

or

This matches the form where:

Since the given directrix is , we equate it to the directrix :

Substituting the values of and in terms of :

Solving this equation will give us the value of :

1. Substitute and into the equation:
3. Multiply the entire equation by to eliminate the denominators:
5. Simplify:
7. Reorder the equation:

This is a quadratic equation in the form .

We use the quadratic formula, where , , and :

1. Calculate the two possible values for :

Among the given options, one of the values of that matches is .

Therefore, the correct answer is 4.

Given two equations:

We need to find the relationship between and . Let's begin by using an algebraic identity for the sum of cubes:

The identity for the sum of cubes is:

By letting and , we use this identity to express:

Now substitute the given :

Note that (Pythagorean identity). Hence:

Therefore, can be rewritten as:\)

Given that , we substitute:

From the square of the first equation, we have:

\)

Using , we find:

Solving for gives:

Substitute this back into the expression for :

Simplify this expression:

This simplifies to: \)

Expanding this gives:

Rewriting yields the equation:

Thus, the correct answer is: