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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

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3 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
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3 years ago
```							 Let PQ be the normal chord to the parabola y2=4x where a=1Let (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 isy−2t1​=2−2t1​​(x−t12​)⇒y−2t1​=−t1​(x−t12​) …………(1)Since Q is a point at the normal, substituting y=2t2​ and x=t22​we have,2t2​−2t1​=−t1​(t22​−t12​)⇒2(t2​−t1​)=−t1​(t2​+t1​)(t2​−t1​)⇒2=−t1​(t2​+t1​)……… (2)It is given that the normal chord subtends a right angle at the vertex A(0,0) i.e., PA⊥AQ∴ (slope of PA)(slope of AQ)=−1((t12​−0)(2t1​−0)​)(t22​−02t2​−0​)=−1⇒t12​2t1​​⋅t22​2t2​​=−1⇒t1​t2​4​=−1⇒t1​t2​=−4 ……(2)From (2),−t1​(t2​+t1​)=2⇒−t1​t2​−t12​=2⇒4−t12​=2 [from (3)]⇒t12​=2⇒t1​=2​Substituting value of t1​ in (3) we havet2​=2​−4​t2​=2−42​​=−22​∴ coordinates of P=((2​)2,2(2​))=(2,22​)Coordinates of Q=((−22​)2,2(−22​))=(−8,−42​)∴ Length of normal chord PQ=(8−2)2+(−42​−22​)2​=62+(−62​)2​=36+72​=108​=63​.
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2 months ago
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