Multiple choice

If $\left( \alpha ,\beta \right) $ is a point on the chord $PQ$ of the circle ${ x }^{ 2 }+{ y }^{ 2 }=19,$ where the coordinate of $P$ and $Q$ are $(3,-4)$ and $(4,3)$ respectively, then

  1. $\alpha \in \left[ 3,4 \right] ,\beta \in \left[ -4,3 \right] $
  2. $\alpha \in \left[ -4,3 \right] ,\beta \in \left[ 4,3 \right] $
  3. $\alpha \in \left[ 3,3 \right] ,\beta \in \left[ -4,4 \right] $

  4. None of these
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A Correct answer
Explanation

The chord PQ connects (3, -4) and (4, 3). Any point (alpha, beta) on the chord must have its coordinates bounded by the coordinates of the endpoints P and Q. Thus, alpha must be between 3 and 4, and beta must be between -4 and 3.

AI explanation

The point (alpha, beta) lies on the line segment joining P(3, -4) and Q(4, 3). Therefore, the x-coordinate alpha must lie between the x-coordinates of P and Q, giving alpha in the interval [3, 4]. Similarly, the y-coordinate beta must lie between the y-coordinates of P and Q, giving beta in the interval [-4, 3]. This matches the condition alpha in [3, 4] and beta in [-4, 3].

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