Multiple choice

If a chord of the circle ${ x }^{ 2 }+{ y }^{ 2 }=8$ makes equal intercepts of length $a$ on the coordinate axes, then

  1. $\left| a \right| <8$
  2. $\left| a \right| <4\sqrt { 2 } $
  3. $\left| a \right| <4$
  4. $\left| a \right| >4$
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C Correct answer
Explanation

A chord making equal intercepts 'a' on axes means the line equation is x/a + y/a = 1, or x + y = a. The distance from the origin (0,0) to the line x + y - a = 0 must be less than the radius sqrt(8). Distance = |a| / sqrt(1^2 + 1^2) = |a| / sqrt(2). So, |a| / sqrt(2) < sqrt(8) implies |a| < sqrt(16) = 4.

AI explanation

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

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