Multiple choice

A chord of the circle $x^{2} + y^{2} = 8$ makes equal intercepts on the coordinate axes. If the length of each intercept is '$a$' then the range of '$a$' is

  1. $(-2, 2)$
  2. $(-4, 4)$
  3. $(-2\sqrt{2},2\sqrt{2})$
  4. $( -1, 1)$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The chord makes equal intercepts 'a' on axes, so it lies on x + y = a or x - y = a. Distance from origin to x + y - a = 0 must be less than radius sqrt(8). Distance = |a| / sqrt(2) < sqrt(8) = 2 * sqrt(2). So |a| < 4, meaning -4 < a < 4.

AI explanation

A line making equal intercepts of length a on the axes has the equation x + y = a or x - y = a. For this line to be a chord of the circle x^2 + y^2 = 8, the perpendicular distance from the origin (0,0) to the line must be less than the radius, which is sqrt(8) or 2*sqrt(2). The perpendicular distance from the origin to x + y - a = 0 is |a| / sqrt(1^2 + 1^2), so we have |a| / sqrt(2) < 2*sqrt(2). Multiplying both sides by sqrt(2) gives |a| < 4, which means the range of a is (-4, 4).

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