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

Answer 1

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,

y−k=− 2 h ( x−k )

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

2−k=− 2 h ( 1−k ) k=2+ 2 h ( 1−h ) (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 ( 1−h ) h 3 4 =2h+2−2h h 3 =8 h=2

So, the value of k will be,

k= 2 2 4 k=1

Thus, the equation of normal is,

y−1= −2 2 ( x−2 ) y−1=−1( x−2 ) x+y=3

Therefore, the correct option is (A).