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Question

The normal to the curve x 2 = 4 y passing (1, 2) is (A) x + y = 3 (B) x − y = 3 (C) x + y = 1 (D) x − y = 1

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Solution

The given equation of curve is,

x 2 =4y

Differentiate the given equation with respect to x,

2x=4 dy dx dy dx = x 2

The slope of normal to the point ( h,k ) is,

1 dy dx | ( h,k ) = 1 h 2 = 2 h

The equation of normal to the given curve at the point ( h,k ) is,

yk= 2 h ( xk )

Since, the normal passes through the point ( 1,2 ), then, the equation of normal will be,

2k= 2 h ( 1k ) k=2+ 2 h ( 1h ) (1)

As the point ( h,k ) lies on the curve x 2 =4y, then,

h 2 =4k k= h 2 4

Substitute the value of k in equation (1),

h 2 4 =2+ 2 h ( 1h ) h 3 4 =2h+22h h 3 =8 h=2

So, the value of k will be,

k= 2 2 4 k=1

Thus, the equation of normal is,

y1= 2 2 ( x2 ) y1=1( x2 ) x+y=3

Therefore, the correct option is (A).


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