If the curves \(y^2 = 6x\), \(9x^2 + by^2 = 16\) intersect each other at right angles, then the value of b is:
(A) 6
(B) 7
(C) 2
(D) 4
(E) 9
(F) 2
Answer (1)
To find the value of b for which the curves y 2 = 6 x and 9 x 2 + b y 2 = 16 intersect at right angles, we will use the concept of orthogonal curves. Two curves intersect orthogonally if at the point of intersection, the product of their slopes is − 1 .
Step 1: Find the slope of the first curve.
The curve y 2 = 6 x can be differentiated with respect to x to find its slope:
d x d ( y 2 ) = d x d ( 6 x )
2 y d x d y = 6
d x d y = y 3
This slope is the derivative d x d y of the first curve at any point ( x , y ) .
Step 2: Find the slope of the second curve.
Now, consider the curve 9 x 2 + b y 2 = 16 . Differentiate this curve with respect to x :
d x d ( 9 x 2 ) + d x d ( b y 2 ) = d x d ( 16 )
18 x + 2 b y d x d y = 0
2 b y d x d y = − 18 x
d x d y = − b y 9 x
This slope is the derivative d x d y of the second curve at any point ( x , y ) .
Step 3: Set up the orthogonality condition.
For the curves to intersect orthogonally, the product of their slopes must be − 1 :
( y 3 ) ( − b y 9 x ) = − 1
Simplifying this expression:
− b y 2 27 x = − 1
b y 2 27 x = 1
Step 4: Use the point of intersection from the first curve.
Since the point ( x , y ) lies on the first curve y 2 = 6 x , we substitute y 2 = 6 x into the equation:
b ( 6 x ) 27 x = 1
6 b 27 = 1
6 27 = b
4.5 = b
Since none of the given options match b = 4.5 , there might be an error in the options listed, but according to the steps, b = 4.5 .
Any discrepancies in provided multiple-choice options should be double-checked with the problem source or further context.
