Question:
The mirror image of any point on the directrix of $ y^2 = 4(x + 1)$ lies on
The mirror image of any point on the directrix of $ y^2 = 4(x + 1)$ lies on
Updated On: Jul 7, 2022
- $3x + 4y- 16 = 0 $
- $3x - 4y +16 = 0 $
- $3x + 4y + 16 = 0 $
- $3x - 4y - 16 = 0.$
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The Correct Option is B
Solution and Explanation
Directrix of $y^{2}= 4\left(x+1\right)$ is $x=-2$
Any point on it is $\left(-2, k\right)$
Now mirror image (x, y) of (- 2, k) in the line x + 2y = 3 is given by
$Also y = k +4 -\frac{8k}{5}$
y = k +4 - $\frac{8k}{5}$
Hence y = 4 - $\frac{3k}{5}=4+\frac{3}{5}\left(\frac {5x}{4}\right)=\frac {16+3x}{4}$
$\Rightarrow\,\,$3x -4y + 16 = 0 is the reqd. mirror image.
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Notes on Parabola
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.