Multiple choice

A circle of radius 4, drawn on a chord of the parabola $y^2\, =\, 8x$ as diameter, touches the axis of the parabola. Then, the slope of the chord is

  1. $\displaystyle \frac{1}{2}$
  2. $\displaystyle \frac{3}{4}$
  3. $1$
  4. $2$
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C Correct answer
AI explanation

The endpoints of the chord on the parabola y^2 = 8x are t1 = (2t1^2, 4t1) and t2 = (2t2^2, 4t2). The equation of the diameter circle is (x - 2t1^2)(x - 2t2^2) + (y - 4t1)(y - 4t2) = 0, and setting x = 0 because it touches the parabola's axis yields y^2 - y(4t1 + 4t2) + 16t1t2 = 0 having exactly one root. This requires the discriminant to be zero, so 16(t1 + t2)^2 - 64t1t2 = 0, simplifying to t1^2 + t2^2 - 2t1t2 = 0, meaning t1 = t2. The radius is given as 4, so applying the distance formula root((2t1^2 - 2t2^2)^2 + (4t1 - 4t2)^2) = 4 simplifies to 4|t1 - t2| = 4, which means t1 - t2 = 1. The slope of the chord is (4t1 - 4t2) / (2t1^2 - 2t2^2) = 2 / (t1 + t2), and substituting t1 = t2 + 1 gives a slope of 1.

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