Multiple choice

The equation of the chord of the circle ${x}^{2}+{y}^{2}={a}^{2}$ which cuts the circle at two points which lie at a distance $d$ from a point $A(\alpha,\beta)$ on the circle is $\alpha x+\beta y=\lambda$, then $\lambda=$

  1. ${a}^{2}-{d}^{2}$
  2. $\cfrac{{a}^{2}}{2}-{d}^{2}$
  3. ${a}^{2}-\cfrac{{d}^{2}}{2}$
  4. ${a}^{2}+\cfrac{{d}^{2}}{2}$
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A Correct answer
AI explanation

Because the chord is at a distance d from point A on the circle, the perpendicular from A to the chord forms a right triangle with a leg of length d. Let the point of intersection of the perpendicular be M, so AM = d and the distance OM from the center O is given by OM^2 = a^2 - d^2. The line passes through the midpoint M, whose coordinates involve alpha and beta, leading to the condition lambda = alpha^2 + beta^2 - d^2. Since the point A(alpha, beta) lies on the circle, alpha^2 + beta^2 = a^2, substituting this yields lambda = a^2 - d^2.

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