Multiple choice

The locus of the middle points of the chords of the circle $\mathrm{x}^{2}+\mathrm{y}^{2}-4\mathrm{x}-2\mathrm{y}-40=0$ which make an angle $\displaystyle \tan^{-1}\frac{4}{3}$ with the $\mathrm{x}$-axis is

  1. $4x-3y =5$
  2. $3\mathrm{x}+4\mathrm{y}=10$
  3. $4\mathrm{x}+3\mathrm{y}=11$
  4. $3\mathrm{x}-4\mathrm{y}=1$
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B Correct answer
Explanation

The center of the circle is (2, 1). The slope of the chord is tan(tan^-1(4/3)) = 4/3. The slope of the line connecting the center to the midpoint of the chord is the negative reciprocal of the chord's slope, which is -3/4. The equation of the line passing through (2, 1) with slope -3/4 is y - 1 = -3/4(x - 2) => 4y - 4 = -3x + 6 => 3x + 4y = 10.

AI explanation

The center of the circle x squared plus y squared minus 4x minus 2y minus 40 equals 0 is (2, 1). A chord making an angle whose tangent is 4/3 with the x-axis has a slope of 4/3, and the locus of the middle points of such chords is perpendicular to the chords and passes through the center. The slope of this locus is the negative reciprocal of 4/3, which is minus 3/4, so using the point-slope form through (2, 1) gives y minus 1 equals minus 3/4 times (x minus 2). Multiplying by 4 and rearranging yields the equation 3x plus 4y equals 10.

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