# Find the Vertex, Focus, Axis, Directrix and Latus-rectum of the Following Parabola Y2 = 5x − 4y − 9 - Mathematics

Find the vertex, focus, axis, directrix and latus-rectum of the following parabola

y2 = 5x − 4y − 9

#### Solution

Given:

y 2 = 5x − 4y − 9

$\Rightarrow y^2 + 4y = 5x - 9$
$\Rightarrow \left( y + 2 \right)^2 = 5x - 5 = 5\left( x - 1 \right)$

Putting $Y = y + 2$

$X = x - 1$

$Y^2 = 5X$

Comparing the given equation with $Y^2 = 4aX$

$4a = 5 \Rightarrow a = \frac{5}{4}$

∴ Vertex = (= 0, = 0) =$\left( x = 1, y = - 2 \right)$

Focus = (aY = 0) =$\left( x - 1 = \frac{5}{4}, y + 2 =0 \right) = \left( x = \frac{9}{4}, y = - 2 \right)$

Equation of the directrix:
X = −a
i.e.$x - 1 = \frac{- 5}{4} \Rightarrow x = \frac{- 1}{4}$

Axis = Y = 0
i.e.$y + 2 = 0 \Rightarrow y = - 2$

Length of the latus rectum = 4a = 5 units

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#### APPEARS IN

RD Sharma Class 11 Mathematics Textbook
Chapter 25 Parabola
Exercise 25.1 | Q 4.8 | Page 24