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)

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This interesting question is from Calculus Application in division Application of Calculus of Mathematics – Class 12

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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).