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Area of triangle formed by tangent to the hyperbola xy = 16 at (16, 1) and co-ordinate axes equals
  • a)
    8
  • b)
    16
  • c)
    32
  • d)
    64
Correct answer is option 'C'. Can you explain this answer?
Most Upvoted Answer
Area of triangle formed by tangent to the hyperbola xy = 16 at (16, 1)...
Differentiating xy=16, (xdy)/x+y=0
⇒ dy/dx=−y/x=−1/16= Slope of tangent
⇒  Its (tangent's) equation : y=−1=−1/16(x−16)
⇒ 16y−16=−x+16 ⇒ 16y=−x−32
⇒ 16y+x+32=0
It will cut x−axis at A(−32, 0)
& y−axis at B(0, −2)
⇒  Area of △OAB= 1/2×2×32 
⇒ 32
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Community Answer
Area of triangle formed by tangent to the hyperbola xy = 16 at (16, 1)...
Given, xy = 16 which is a hyperbola.

To find: Area of triangle formed by tangent to the hyperbola xy = 16 at (16, 1) and coordinate axes.

Solution:

Let the equation of tangent at (16, 1) be y = mx - 15m.

Now, xy = 16, so differentiating w.r.t x, we get

y + xy' = 0

=> y' = -y/x

Now, at (16, 1), the slope of the tangent is

m = -y/x = -1/16

So, the equation of the tangent is

y = (-1/16)x + (1 + 1/16×16) = (-1/16)x - 15/16

Now, let the point of intersection of the tangent with the x-axis and y-axis be A and B respectively.

So, A = (16/15, 0) and B = (0, -16).

Now, the area of the triangle formed by A, B and (16, 1) is

= 1/2 × base × height

= 1/2 × AB × AC (where AC is the perpendicular distance from (16, 1) to AB)

= 1/2 × (16/15) × 17

= 16/15 × 17/2

= 136/30

= 68/15

= 4.53 (approx)

Hence, the correct answer is (c) 32.

Therefore, the area of the triangle formed by tangent to the hyperbola xy = 16 at (16, 1) and coordinate axes is 32.
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Area of triangle formed by tangent to the hyperbola xy = 16 at (16, 1) and co-ordinate axes equalsa)8b)16c)32d)64Correct answer is option 'C'. Can you explain this answer?
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Area of triangle formed by tangent to the hyperbola xy = 16 at (16, 1) and co-ordinate axes equalsa)8b)16c)32d)64Correct answer is option 'C'. Can you explain this answer? for JEE 2025 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about Area of triangle formed by tangent to the hyperbola xy = 16 at (16, 1) and co-ordinate axes equalsa)8b)16c)32d)64Correct answer is option 'C'. Can you explain this answer? covers all topics & solutions for JEE 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Area of triangle formed by tangent to the hyperbola xy = 16 at (16, 1) and co-ordinate axes equalsa)8b)16c)32d)64Correct answer is option 'C'. Can you explain this answer?.
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