If is the directrix of the parabola , then is:

4 Explanation: Solution 1. Concept: The directrix of a parabola whose equation is of the form is the line . 2. Rearranging the Given Equation: The given equation of the parabola is . We can rewrite this to match the general form: 3. Identifying Constants: By comparing equation with , we get: 4. Using the Directrix Equation: The equation of the directrix is . According to the question, the directrix is . Therefore: 5. Solving for : Multiply the entire equation by to clear the denominators: Factor the quadratic equation: This gives two possible values for : Based on the given options, is the correct choice. Correct Option: 3 ( )

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NIMCET 2014 — Mathematics PYQ

NIMCET | Mathematics | 2014

If $x = 1$ is the directrix of the parabola $y^2 = kx - 8$ , then $k$ is:

Choose the correct answer:

B.
8

C.
4
(Correct Answer)

D.
$\frac{1}{4}$

Explanation

Solution

1. Concept:

The directrix of a parabola whose equation is of the form $(y - q)^2 = 4a(x - p)$ is the line $x = p - a$ .

2. Rearranging the Given Equation:

The given equation of the parabola is $y^2 = kx - 8$ . We can rewrite this to match the general form:

$(y - 0)^2 = k\left(x - \frac{8}{k}\right)$

$(y - 0)^2 = 4\left(\frac{k}{4}\right)\left(x - \frac{8}{k}\right) \quad \dots(1) \text{}$

3. Identifying Constants:

By comparing equation $(1)$ with $(y - q)^2 = 4a(x - p)$ , we get:

$q = 0$

$a = \frac{k}{4}$

$p = \frac{8}{k}$

4. Using the Directrix Equation:

The equation of the directrix is $x = p - a$ .

$x = \frac{8}{k} - \frac{k}{4} \quad \dots(2) \text{}$

According to the question, the directrix is $x = 1$ . Therefore:

$\frac{8}{k} - \frac{k}{4} = 1 \text{}$

5. Solving for $k$ :

Multiply the entire equation by $4k$ to clear the denominators:

$32 - k^2 = 4k \text{}$

$k^2 + 4k - 32 = 0 \text{}$

Factor the quadratic equation:

$k^2 + 8k - 4k - 32 = 0 \text{}$

$k(k + 8) - 4(k + 8) = 0 \text{}$

$(k + 8)(k - 4) = 0 \text{}$

This gives two possible values for $k$ :

$k + 8 = 0 \Rightarrow k = -8$

$k - 4 = 0 \Rightarrow k = 4$

Based on the given options, $k = 4$ is the correct choice.

Correct Option: 3 ( $4$ )