Find the vertex, axis, focus, directrix, tangent at the vertex, and length of the latus rectum of the parabola:

2y²+ 3y - 4x - 3 = 0

Question

Find the vertex, axis, focus, directrix, tangent at the vertex,
and length of the latus rectum of the parabola:
2y2 + 3y - 4x - 3 = 0

Lalit Mehra · Accepted Answer

The equation of the parabola is 2y2 + 3y â€“ 4x â€“ 3 = 0 $2 y^{2} + 3 y - 4 x - 3 = 0 ∴ 2 y^{2} + 3 y = 4 x + 3 \Rightarrow y^{2} + \frac{3}{2} y = 2 x + \frac{3}{2} \Rightarrow y^{2} + \frac{3}{2} y + \frac{9}{16} = 2 x + \frac{3}{2} + \frac{9}{16} \Rightarrow (y + \frac{3}{4})^{2} = 2 x + \frac{33}{16} \Rightarrow (y + \frac{3}{4})^{2} = 2 (x + \frac{33}{32}) Let y + \frac{3}{4} = Y and x + \frac{33}{32} = X ∴ Y^{2} = 2 X$ This is of the form Y2 = 4ax, where 4a = 2 4a = 2 $∴ a = \frac{1}{2}$ Vertex is given by X = 0 and Y = 0 $∴ x + \frac{33}{32} = 0 and y + \frac{3}{4} = 0 \Rightarrow x = - \frac{33}{32} and y = - \frac{3}{4}$ Vertex of the parabola = $(- \frac{33}{32}, - \frac{3}{4})$ Axis is given by Y = 0 $∴ y + \frac{3}{4} = 0$ Axis of the parabola is $y + \frac{3}{4} = 0$ Focus is given by $X = \frac{1}{2} and Y = 0$ $x + \frac{33}{32} = \frac{1}{2} and y + \frac{3}{4} = 0 ∴ x = \frac{1}{2} - \frac{33}{32} and y = - \frac{3}{4} \Rightarrow x = \frac{16 - 33}{32} and y = - \frac{3}{4} \Rightarrow x = - \frac{17}{32} and y = - \frac{3}{4}$ Focus of the parabola $= (- \frac{17}{32}, - \frac{3}{4})$ Equations of directrix is given by $X = - \frac{1}{2}$ $∴ x + \frac{33}{32} = - \frac{1}{2} \Rightarrow x = - \frac{1}{2} - \frac{33}{32} = \frac{- 16 - 33}{32} = - \frac{49}{32} \Rightarrow x + \frac{49}{32} = 0$ Equation of the directrix is $x + \frac{49}{32} = 0$ Equation of tangent at the vertex is given by X = 0 $∴ x + \frac{33}{32} = 0$ Equation of tangent at the vertex is $x + \frac{33}{32} = 0$ Length of the latus rectum = 4a = 2

Find the vertex, axis, focus, directrix, tangent at the vertex, and length of the latus rectum of the parabola: 2y2 + 3y - 4x - 3 = 0

Find the vertex, axis, focus, directrix, tangent at the vertex, and length of the latus rectum of the parabola:

2y²+ 3y - 4x - 3 = 0