Department of Pre-University Education, KarnatakaPUC Karnataka Science Class 11

# Find the Vertex, Focus, Axis, Directrix and Latus-rectum of the Following Parabolas Y2 − 4y − 3x + 1 = 0 - Mathematics

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

y2 − 4y − 3x + 1 = 0

#### Solution

Given:
y2 − 4y − 3x + 1 = 0

$\Rightarrow \left( y - 2 \right)^2 - 4 - 3x + 1 = 0$
$\Rightarrow \left( y - 2 \right)^2 = 3\left( x + 1 \right)$
$\Rightarrow \left( y - 2 \right)^2 = 3\left( x - \left( - 1 \right) \right)$

Let $Y = y - 2$

$X = x + 1$

Then, we have:

$Y^2 = 3X$

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

$4a = 3 \Rightarrow a = \frac{3}{4}$

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

Focus = (X = a= 0) = $\left( x + 1 = \frac{3}{4}, y - 2 = 0 \right) = \left( x = \frac{- 1}{4}, y = 2 \right)$

Equation of the directrix:
X = −a
i.e$x + 1 = \frac{- 3}{4} \Rightarrow x = \frac{- 7}{4}$

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

Length of the latus rectum = 4a = 3 units

Is there an error in this question or solution?

#### APPEARS IN

RD Sharma Class 11 Mathematics Textbook
Chapter 25 Parabola
Exercise 25.1 | Q 4.3 | Page 24