Multiple choice

For the circle C with the equation $x^2 \ + \, y^2 \, - \, 16x \, - \, 12y \, + \, 64 \, = \, 0$ match the list-I with the list - II given below : List - I List -II i The equation of the polar of (-5, 1) with respect to C a y = 0 ii The equation of the tangent at (8, 0) to C b y = 6 iii The equation of the normal at (2, 6) to C c x + Y = 7 iv The equation of the diameter of the C through (8, 12) d 13x + 5y = 98 e x = 8

  1. i - d, ii - b, iii - a, iv - c

  2. i - d, ii - a, iii - b, iv - e

  3. i - c, ii - d, iii - a, iv - b

  4. i - c, ii - e, iii - b, iv - a
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B Correct answer
Explanation

The circle equation is (x-8)^2 + (y-6)^2 = 36. i: Polar of (-5, 1) is (x+5)(8) + (y-1)(6) - 8(x-5) - 6(y+1) + 64 = 0, which simplifies to 13x + 5y = 98. ii: Tangent at (8, 0) is (x-8)(0) + (y-6)(-6) = 36, which is y=0. iii: Normal at (2, 6) is y=6. iv: Diameter through (8, 12) is x=8.

AI explanation

The circle center is (8, 6) with radius 6. Using the polar formula xx1 plus yy1 minus 8x minus 8x1 minus 6y minus 6y1 plus 64 equals 0 for the point (-5, 1) yields 13x plus 5y equals 98. The tangent at (8, 0) is the horizontal line y equals 0. The normal at (2, 6) is the horizontal line y equals 6. The diameter through (8, 12) is the vertical line x equals 8.

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