# P Find the Vertex, Focus, Axis, Directrix and Latus-rectum of the Following Parabola 4 (Y − 1)2 = − 7 (X − 3) - Mathematics

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

4 (y − 1)2 = − 7 (x − 3)

#### Solution

Given:

4(y − 1)2 = − 7 (x − 3)

$\Rightarrow \left( y - 1 \right)^2 = \frac{- 7}{4}\left( x - 3 \right)$

Let $Y = y - 1$

$X = x - 3$

Then, we have:

$Y^2 = \frac{- 7}{4}X$

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

$4a = \frac{7}{4} \Rightarrow a = \frac{7}{16}$

∴ Vertex = (X = 0, = 0) = $\left( x = 3, y = 1 \right)$

Focus = (X = −a, Y = 0) = $\left( x - 3 = \frac{- 7}{16}, y - 1 = 0 \right) = \left( x = \frac{41}{16}, y = 1 \right)$

Equation of the directrix:

X = a
i.e. $x - 3 = \frac{7}{16} \Rightarrow x = \frac{55}{16}$

Axis = Y = 0
i.e. $y - 1 = 0 \Rightarrow y = 1$

Length of the latus rectum = 4a = $\frac{7}{4}$

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#### APPEARS IN

RD Sharma Class 11 Mathematics Textbook
Chapter 25 Parabola
Exercise 25.1 | Q 4.7 | Page 24