Question

Equation of directrix of the parabola y^2=5x-4y-9

Answer 1

$\blue{\bold{\underline{\underline{Answer:}}}}$

$\green{\tt{\therefore{x+\frac{1}{4}=0}}}$

$\orange{\bold{\underline{\underline{Step-by-step\:explanation:}}}}$

$\green{\underline \bold{Given:}} \\ \tt: \implies {y}^{2} = 5x - 4y - 9 \\ \\ \red{\underline \bold{To \: Find:}} \\ \tt: \implies Eqn \: of \: directrix = ?$

• According to given question :

$\tt: \implies {y}^{2} = 5x - 4y - 9 \\ \\ \tt: \implies {y}^{2} + 4y = 5x - 9 \\ \\ \tt: \implies {y}^{2} + 4y + 4 = 5x - 9 + 4 \\ \\ \tt: \implies {(y - 2)}^{2} = 5x - 5 \\ \\ \tt: \implies {(y - 2)}^{2} = 5(x - 1) \\ \\\tt: \implies {(y - 2)}^{2} = 4 \times \frac{5}{4}(x - 1) \\ \\ \text{So, \: it \: is \: in \: the \: form \: of}\\ \\ \tt: \implies Y^{2} = 4aX \\ \\ \bold{Where : } \\ \tt \circ \: a = \frac{5}{4} \\ \\ \bold{As \: we \: know \: that} \\ \tt: \implies Eqn \: of \: directrix \to X = - a \\ \\ \tt: \implies Eqn \: of \: directrix \to x - 1 = - \frac{5}{4} \\ \\ \tt: \implies Eqn \: of \: directrix \to x = - \frac{5}{4} + 1 \\ \\ \tt: \implies Eqn \: of \: directrix \to x = \frac{ - 1}{4} \\ \\ \green{\tt: \implies Eqn \: of \: directrix \to x + \frac{1}{4} = 0}$

Answer 2

Answer:

$\large\boxed{\sf{x=-\dfrac{1}{4}}}$

Step-by-step explanation:

Given an equation of parabola such that,

${y}^{2} = 5x - 4y - 9$

To find the equation of directrix.

We know that, general form of a parabola is:

• ${Y}^{2}=4AX$

Therefore, we need to convert the given eqn in this form.

Thus, further solving, we get,

$= > {y}^{2} + 4y = 5x - 9\\ \\ = > {y}^{2} + 4y + 4 = 5x - 9 + 4 \\ \\ = > {(y)}^{2} + 2(y)(2) + {(2)}^{2} = 5x - 5 \\ \\ = > {(y + 2)}^{2} = 5(x - 1) \\ \\ = > {(y + 2)}^{2} = 4 \times 5(x - 1) \times \dfrac{1}{4} \\ \\ = > {(y + 2)}^{2} = 4 ( \dfrac{5}{4}) (x - 1)$

On comparing the equations, we get,

• Y = (y + 2)
• A = 5/4
• X = (x - 1)

Now, we know that,

General equation for directrix is :

• X = -A

Therefore, substituting the values, we get,

$= > x - 1 = - \dfrac{5}{4} \\ \\ = > 4(x - 1) = - 5 \\ \\ = > 4x - 4 = - 5 \\ \\ = > 4x - 4 + 5 = 0 \\ \\ = > 4x + 1 = 0 \\ \\ = > 4x = - 1 \\ \\ = > x = - \dfrac{1}{4}$

Hence, required eqn of directrix is x = - ¼