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Length of a normal chord of the parabola,y^2=4x, which makes an angle of pi/4 with the axis of x is.Answer is 8 root 2 :( please help its urgent

Length of a normal chord of the parabola,y^2=4x, which makes an angle of pi/4 with the axis of x is.Answer is 8 root 2 :( please help its urgent

Grade:11

2 Answers

Vaibhav
50 Points
5 years ago
Eq of normal y=mx-am^3-2 am since m=1 a=1 putting value give y=x-3 put y=x-3 in y^2=4x solving gives x=1,9 from y=x-3 get y=-2 , 6 points known then from distance formula you get answers please approved if understand
ankit singh
askIITians Faculty 614 Points
2 years ago

 

Let PQ be the normal chord to the parabola y2=4x where a=1
Let (at12,2at1) and (at22,2at2) be the coordinates of P and Q respectively.
i.e., coordinates of P=(t12,2t1)
and coordinates of Q=(t22,2t2)
Equation of normal at point P is
y2t1=22t1(xt12)
y2t1=t1(xt12) …………(1)
Since Q is a point at the normal, substituting y=2t2 and x=t22
we have,
2t22t1=t1(t22t12)
2(t2t1)=t1(t2+t1)(t2t1)
2=t1(t2+t1)……… (2)
It is given that the normal chord subtends a right angle at the vertex A(0,0) i.e., PAAQ
(slope of PA)(slope of AQ)=1
((t120)(2t10))(t2202t20)=1
t122t1t222t2=1
t1t24=1
t1t2=4 ……(2)
From (2),
t1(t2+t1)=2
t1t2t12=2
4t12=2 [from (3)]
t12=2
t1=2
Substituting value of t1 in (3) we have
t2=24
t2=242=22
coordinates of P=((2)2,2(2))=(2,22)
Coordinates of Q=((22)2,2(22))=(8,42)
Length of normal chord PQ=(82)2+(4222)2
=62+(62)2
=36+72
=108
=63.

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