Multiple choice

If the common chord of the circle ${x^2} + {(y - \lambda )^2} = 16$ and $x^2 + y^2 = 16$ subtend a right angle at the origin then $\lambda$ is equal to

  1. 4
  2. $4\sqrt{2}$
  3. $\pm 4\sqrt{2}$

  4. 8
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The common chord of x^2 + (y-lambda)^2 = 16 and x^2 + y^2 = 16 is found by subtracting the equations: x^2 + y^2 - 2y*lambda + lambda^2 = 16 and x^2 + y^2 = 16. This yields -2y*lambda + lambda^2 = 0, so y = lambda/2. The chord is a horizontal line. For it to subtend 90 degrees at the origin, the distance from the origin to the chord must be related to the radius.

AI explanation

To find the common chord, subtract the second circle's equation from the first to get y = lambda/2. Substituting y = lambda/2 back into the circle x^2 + y^2 = 16 yields x^2 + (lambda/2)^2 = 16, which simplifies to x = plus or minus sqrt(16 - lambda^2/4). The endpoints of the common chord at the origin form a right angle, meaning the product of their slopes is -1. The slopes are sqrt(16 - lambda^2/4) / (lambda/2) and -sqrt(16 - lambda^2/4) / (lambda/2), so setting their product to -1 gives (16 - lambda^2/4) / (lambda^2/4) = 1. Solving this results in 64 - lambda^2 = lambda^2, so lambda^2 = 32 and lambda is plus or minus 4*sqrt(2). The correct option is C.

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