If ax + by = c is tangent to the circle x2 + y2 = 16, then which of the following is correct?

  1. 16 (a2 + b2) = c2
  2. 16 (a2 - b2) = c2
  3. 16 (a2 + b2) = -c2
  4. 16 (a2 - b2) = -c2

Answer (Detailed Solution Below)

Option 1 : 16 (a2 + b2) = c2
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Detailed Solution

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Concept:

Tangent to a Circle: The equation of the tangent to a circle x2 + y2 = r2 at a point (a, b) is given by ax + by = r2.

Calculation:

Comparing the equation x2 + y2 = 16 with the general equation of the circle i.e. x2 + y2 = r2, we have r2 = 16.

Also, since the line ax + by = r2 is tangent to x2 + y2 = r2 at the point (a, b) on the circle, we can say that for the line ax + by = c to be a tangent to x2 + y2 = 16, (a, b) must be on the circle and c = r2 = 16.

∴ a2 + b2 = c

⇒ c(a2 + b2) = c2

16(a2 + b2) = c2.

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