Question

if variable chord y=mx+c to curve x²-4y=0 always subtends right angle at the origin, then
A) c=2
B) c=4
C) c=8
D) none of these​

Answer 1

Given :  variable chord y=mx+c to curve x²-4y=0 always subtends right angle at the origin,

To find : Value of c  

A) c=2

B) c=4

C) c=8

D) none of these​

Solution:

x²  -  4y=0

x = 2a   ,   y = a²

x = 2b   ,   y  = b²

Let say chord y = mx + c   has end point

(2a , a² )  , ( 2b  , b² )

Slope at origin  =  a²/2a = a/2  and  b/2

(a/2)(b/2) = - 1

=> ab = - 4

=> b = - 4/a

y = mx + c

=>  a² = 2am  + c    ,   b² = 2bm  + c

a² - b²  = 2m(a - b)

=> a + b  =  2m

Substitute  b = - 4/a

a  - 4/a =  2m  

=> a²  - 4  = 2am

=> a² = 2am + 4

compare with

a² = 2am  + c  

=> c = 4

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