NIMCET 2014 — Mathematics PYQ

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Question

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:

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 ( $4$ )