Question:
The normal chord at a point 't' on the parabola $ y^2 = 4ax$ subtends a right angle at the vertex. Then $t^2$ =
The normal chord at a point 't' on the parabola $ y^2 = 4ax$ subtends a right angle at the vertex. Then $t^2$ =
Updated On: Jul 7, 2022
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The Correct Option is B
Solution and Explanation
Normal at $'t'$ for parabola $ y^2 = 4ax$ is
$tx + y = 2ax + ax^3 \quad...(1)$
Combined equation of the lines joining the vertex i.e., origin to the pts. of intersection of the parabola and $(1)$ is
$y^{2} = 4ax\left(\frac{tx+y}{2at+at^{3}}\right) $
$ \Rightarrow \left(2t+t^{3}\right)y^{2}=4x\left(y+tx\right) $
Since $\left(1\right)$ makes a right angle at the vertex
$\therefore 2t+ t^{3} - 4t = 0$
$\Rightarrow t^{2} = 2$
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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.