Multiple choice

On a coordinate plane, the circle x2 + y2 = 6ux + 8vy intersects the x-axis at A(α, 0) and the y-axis at B(0, β), where αβ ≠ 0. If point C(p, q) lies on the chord AB, what is the value of p 3 u + q 4 v ?

  1. 1

  2. 2

  3. 6

  4. 4
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The nonzero intercepts are A(6u, 0) and B(0, 8v). The chord equation is x/(6u) + y/(8v) = 1, so p/(3u) + q/(4v) = 2.

AI explanation

Rewrite the circle equation as x squared minus 6ux plus y squared minus 8vy equals zero. To find the x-intercept, set y to zero to get x squared minus 6ux equals zero, meaning x(x - 6u) = 0, so the point A has coordinates (6u, 0). Similarly, setting x to zero gives y squared minus 8vy equals zero, meaning y(y - 8v) = 0, so the point B has coordinates (0, 8v). The equation of the line passing through A and B is x divided by 6u plus y divided by 8v equals 1. Since the point C(p, q) lies on this line chord, substituting p for x and q for y gives p divided by 6u plus q divided by 8v equals 1. Multiplying the entire equation by 2 results in p divided by 3u plus q divided by 4v equals 2.

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