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Chapter 15: Problem 1388

If the line \(\mathrm{x}-1=0\) is the directrix of the parabola \(\mathrm{y}^{2}-\mathrm{kx}+8=0\) then one of the values of \(\mathrm{k}\) is (a) 4 (b) \((1 / 8)\) (c) \((1 / 4)\) (d) 8

Short Answer

Expert verified
One of the values of k is 4, which corresponds to option (a).

Step by step solution

01

1. Determine the general form of a parabola with a vertical axis of symmetry.

The general form of a parabola with a vertical axis of symmetry can be written as: \[y^2 = 4ax\]
02

2. Write the given parabola in vertex form.

The given equation of the parabola is: \[y^2 - kx + 8 = 0\] To write the given equation in vertex form, we need to make it look like the general form. So, we can rewrite the equation as: \[y^2 = kx - 8\]
03

3. Determine the vertex and focus of the parabola.

The vertex form of the given parabola is: \[y^2 = kx - 8\] Comparing the given equation with the general equation, we get 4a = k. The vertex of the parabola is at the origin (0,0), and the distance from the vertex to the focus is a. Therefore, the focus of the parabola is at (a, 0).
04

4. Use the directrix' equation to find the distance between the vertex and the focus.

The given equation of the directrix is x - 1 = 0, or x = 1. The distance from the vertex to the directrix is the same as the distance from the vertex to the focus. In this case, it is 1 unit since the vertex is located at the origin.
05

5. Use the distance between the vertex and the focus to find the value of k.

From the earlier steps, we know that the distance from the vertex to the focus (a) is 1 unit; therefore, 4a = k, and we can substitute a with the distance between the vertex and the focus: \[k = 4(1) = 4\] Hence, one of the values of k is 4. The correct answer is (a) 4.

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Most popular questions from this chapter

Tangents are drawn to the ellipse \(\left(\mathrm{x}^{2} / 9\right)+\left(\mathrm{y}^{2} / 5\right)=1\) at ends of latus recturm line. The area of quadrilateral so formed is \(\ldots \ldots \ldots\) (a) \((27 / 4)\) (b) \((27 / 55)\) (c) 27 (d) \((27 / 2)\)

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