Multiple choice

If the length of tangent drawn from the point $(5,3)$ to the circle $x^2+y^2+2x+ky+17=0$ be $7$, then $k=$

  1. $4$
  2. $-4$
  3. $-6$
  4. $\dfrac{13}{2}$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Length of tangent from (x1, y1) to x^2+y^2+2gx+2fy+c=0 is sqrt(x1^2+y1^2+2gx1+2fy1+c). Here, g=1, f=k/2, c=17. Tangent = 7. 49 = 5^2 + 3^2 + 2(1)(5) + 2(k/2)(3) + 17. 49 = 25 + 9 + 10 + 3k + 17. 49 = 61 + 3k. 3k = -12, k = -4.

AI explanation

Using the standard tangent length formula, the square of the tangent's length equals the power of the point, calculated by substituting the coordinates into the circle's equation. Substituting (5, 3) gives 5^2 + 3^2 + 2(5) + k(3) + 17 = 49. Simplifying this equation yields 25 + 9 + 10 + 3k + 17 = 49, which results in 3k = -12. Solving for k gives a value of -4.

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