At which point does the normal to the hyperbola xy = 4 at (2, 2) intersects the hyperbola again?

Category: QuestionsAt which point does the normal to the hyperbola xy = 4 at (2, 2) intersects the hyperbola again?
Editor">Editor Staff asked 11 months ago

At which point does the normal to the hyperbola xy = 4 at (2, 2) intersects the hyperbola again?
 
(a) (-2, -2)
 
(b) (-2, 2)
 
(c) (2, -2)
 
(d) (0, 2)
 
The question was asked in an online quiz.
 
This interesting question is from Calculus Application in division Application of Calculus of Mathematics – Class 12
NCERT Solutions for Subject Clas 12 Math Select the correct answer from above options 
Interview Questions and Answers, Database Interview Questions and Answers for Freshers and Experience

1 Answers
Editor">Editor Staff answered 11 months ago

Correct answer is (a) (-2, -2)
 
Easy explanation: Equation of the given hyperbola is, xy = 4 ……….(1)
 
Differentiating both side of (1) with respect to y, we get,
 
y*(dx/dy) + x(1) = 0
 
Or dx/dy = -(x/y)
 
Thus, the required equation of the normal to the hyperbola at (2, 2) is,
 
y – 2 = -[dx/dy](2, 2) (x – 2) = -(-2/2)(x – 2)
 
So, from here,
 
y – 2 = x – 2
 
Or x – y = 0 ……….(2)
 
Solving the equation (1) and (2) we get,
 
x = 2 and y = 2 or x = -2 and y = -2
 
Thus, the line (2) intersects the hyperbola (1) at (2, 2) and (-2, -2).
 
Hence, the evident is that the normal at (2, 2) to the hyperbola (1) again intersects it at (-2, -2).