Question

The points on the curve 9 y 2 = x 3 , where the normal to the curve makes equal intercepts with the axes are (A) (B) (C) (D)

Answer 1

The given equation of curve is,

9 y 2 = x 3

Differentiate the given equation with respect to x,

9( 2y ) dy dx =3 x 2 dy dx = x 2 6y

The slope of the normal at the point ( x 1 , y 1 ),

−1 dy dx | ( x 1 , y 1 ) =− 6 y 1 x 1 2

The equation of the normal at point ( x 1 , y 1 ) is,

y− y 1 = −6 y 1 x 1 2 ( x− x 1 ) x 1 2 y− x 1 2 y 1 =−6x y 1 +6 x 1 y 1 6x y 1 + x 1 2 y=6 x 1 y 1 + x 1 2 y 1 6x y 1 6 x 1 y 1 + x 1 2 y 1 + x 1 2 y 6 x 1 y 1 + x 1 2 y 1 =1

Simplify further,

x x 1 ( 6+ x 1 ) 6 + y y 1 ( 6+ x 1 ) x 1 =1

Since the normal makes equal intercepts, then,

x 1 ( 6+ x 1 ) 6 = y 1 ( 6+ x 1 ) x 1 x 1 6 = y 1 x 1 x 1 2 =6 y 1 (1)

Since, the point ( x 1 , y 1 ) lies on the given curve, so,

9 y 1 2 = x 1 3 (2)

From equation (1),

9 ( x 1 2 6 ) 2 = x 1 3 x 1 2 4 = x 1 3 x 1 =4

Substitute the value of x in equation (2),

9 y 1 2 = ( 4 ) 3 9 y 1 2 =64 y 1 2 = 64 9 y 1 =± 8 3

So, the required point is, ( 4,± 8 3 ).

Therefore, the correct option is A.