Question

If point P(4,-2)is one end of focal chord PQ of parabola y²=x , then slope of tangent at Q is

Answer 1

Given : point P(4,-2)is one end of focal chord PQ of parabola y²=x  

To find : Slope of Tangent Q

Solution:

Let say point Q  = ( x, y )

y²=x  

Comparing with

y² =  4ax

=> 4a = 1

=> a  = 1 /4

Hence  Focal chord passes through S ( 1/4 , 0)

Slope of PS  =   (0-(- 2)) /(1/4 - 4) )  =  - 8/15

Slope of PQ = Slope of PS

=> ( y + 2)/(x - 4)  = -8/15

=> 15y + 30  =  -8x + 32

=> 15y =  -8x  + 2

=> 15y  = -8y² + 2

=> 8y² +  15y  - 2 = 0

=> 8y²  + 16y  - y - 2 = 0

=> 8y (y + 2) - 1 (y + 2) = 0

=> (8y - 1)( y + 2) = 0

y = - 2  is P

y =  1/8

=>   y = 1/8   => x  = 1/64

Q = (1/64  , 1/8)

y²=x

=> 2y dy/dx = 1

=> dy/dx = 1/2y

at Q  y = 1/8

=>  dy/dx =  1/2(1/8)

=>  dy/dx =   4

slope of tangent at Q is 4

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