Multiple choice

If ABCD is a convex quadrilateral such that $\displaystyle 4\sec A+5=0$, then the quadratic equation whose roots are $\displaystyle \tan A$ and $\displaystyle co\sec A$ is

  1. $\displaystyle 12x^{2}-29x+15=0$
  2. $\displaystyle 12x^{2}-11x-15=0$
  3. $\displaystyle 12x^{2}+11x-15=0$

  4. none of these
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Given 4sec A + 5 = 0, sec A = -5/4. Since sec A is negative, A is in the second quadrant. Thus, tan A = -3/4 and cosec A = 5/3. The quadratic equation with roots -3/4 and 5/3 is (x + 3/4)(x - 5/3) = 0, which simplifies to x^2 - (5/3 - 3/4)x - 5/4 = 0, or x^2 - (11/12)x - 5/4 = 0. Multiplying by 12 gives 12x^2 - 11x - 15 = 0.

AI explanation

From the given equation 4 sec A + 5 = 0, we find sec A = -5/4, which means cos A = -4/5. Since ABCD is a convex quadrilateral, angle A lies in the second quadrant, making sin A positive and equal to 3/5. Consequently, tan A = sin A / cos A = (3/5) / (-4/5) = -3/4, and cosec A = 1 / sin A = 5/3. For a quadratic equation with roots tan A and cosec A, the sum of the roots is -3/4 + 5/3 = 11/12 and the product is (-3/4)(5/3) = -5/4. The required equation is x^2 - (Sum)x + (Product) = 0, which expands to x^2 - 11/12 x - 5/4 = 0. Multiplying the entire equation by 12 gives 12x^2 - 11x - 15 = 0.

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