Question:
The parametric equation of a parabola is $x = t^2 + 1, y = 2t + 1$. The cartesian equation of its directrix is
The parametric equation of a parabola is $x = t^2 + 1, y = 2t + 1$. The cartesian equation of its directrix is
Updated On: May 11, 2024
- $ y = 0$
- $ x = -1$
- $ x = 0$
- $ x - 1 = 0$
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The Correct Option is C
Solution and Explanation
Given, parametric equation of parabola is $x = t^2 + 1, y = 2t + 1$
$\Rightarrow x = \left(\frac{y -1}{2}\right)^{2} + 1$
$ \Rightarrow \left(y - 1\right)^{2} =4\left(x-1\right) $
$\Rightarrow Y^{2} = 4X $
$ \therefore$ Vertex is (1, 1), length of latus rectum = 4
Clearly, equation of directrix is
$X =- 1 \Rightarrow x-1 =-1\Rightarrow x=0$.
$\Rightarrow x = \left(\frac{y -1}{2}\right)^{2} + 1$
$ \Rightarrow \left(y - 1\right)^{2} =4\left(x-1\right) $
$\Rightarrow Y^{2} = 4X $
$ \therefore$ Vertex is (1, 1), length of latus rectum = 4
Clearly, equation of directrix is
$X =- 1 \Rightarrow x-1 =-1\Rightarrow x=0$.
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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.