Question:
The directrix of the parabola y$^2$ + 4x + 3 = 0 is
The directrix of the parabola y$^2$ + 4x + 3 = 0 is
Updated On: May 12, 2022
- $x-\frac{4}{3}=0$
- $x+\frac{1}{4}=0$
- $x-\frac{3}{4}=0$
- $x-\frac{1}{4}=0$
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The Correct Option is D
Solution and Explanation
The equation of the parabola is
y$^2$ + 4x + 3 = 0
or$\quad\quad\quad y^{2}=-4\left(x+\frac{3}{4}\right)\quad \quad \quad \quad...\left(1\right)$
The directrix of the parabola
$Y^{2}=-4aX\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad... \left(2\right)$
is X = a.
On comparing the equation $\left(1\right)$ and $\left(2\right)$, we
get$\quad\quad4a = 4\quad$ and $X =x+\frac{3}{4}$
or $\quad \quad a = 1\quad$ and $X =x+\frac{3}{4}$
Hence the directrix of the parabola $\left(1\right)$ is
$x+\frac{3}{4}=1 or x-\frac{1}{4}=0.$
y$^2$ + 4x + 3 = 0
or$\quad\quad\quad y^{2}=-4\left(x+\frac{3}{4}\right)\quad \quad \quad \quad...\left(1\right)$
The directrix of the parabola
$Y^{2}=-4aX\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad... \left(2\right)$
is X = a.
On comparing the equation $\left(1\right)$ and $\left(2\right)$, we
get$\quad\quad4a = 4\quad$ and $X =x+\frac{3}{4}$
or $\quad \quad a = 1\quad$ and $X =x+\frac{3}{4}$
Hence the directrix of the parabola $\left(1\right)$ is
$x+\frac{3}{4}=1 or x-\frac{1}{4}=0.$
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Concepts Used:
Parabola
Parabola is defined as the locus of points equidistant from a fixed point (called focus) and a fixed-line (called directrix).
Standard Equation of a Parabola
For horizontal parabola
- Let us consider
- Origin (0,0) as the parabola's vertex A,
- Two equidistant points S(a,0) as focus, and Z(- a,0) as a directrix point,
- P(x,y) as the moving point.
- Let us now draw SZ perpendicular from S to the directrix. Then, SZ will be the axis of the parabola.
- The centre point of SZ i.e. A will now lie on the locus of P, i.e. AS = AZ.
- The x-axis will be along the line AS, and the y-axis will be along the perpendicular to AS at A, as in the figure.
- By definition PM = PS
=> MP2 = PS2
- So, (a + x)2 = (x - a)2 + y2.
- Hence, we can get the equation of horizontal parabola as y2 = 4ax.
For vertical parabola
- Let us consider
- Origin (0,0) as the parabola's vertex A
- Two equidistant points, S(0,b) as focus and Z(0, -b) as a directrix point
- P(x,y) as any moving point
- Let us now draw a perpendicular SZ from S to the directrix.
- Then SZ will be the axis of the parabola. Now, the midpoint of SZ i.e. A, will lie on P’s locus i.e. AS=AZ.
- The y-axis will be along the line AS, and the x-axis will be perpendicular to AS at A, as shown in the figure.
- By definition PM = PS
=> MP2 = PS2
So, (b + y)2 = (y - b)2 + x2
- As a result, the vertical parabola equation is x2= 4by.