Multiple choice

From the point $A(0.3)$ on the circle $x^{2} + 4x + (y - 3)^{2} = 0$ a chord $AB$ is drawn and extended to a point $M$ such that $AM = 2AB$. The equation of the locus of $M$ is

  1. $x^{2} + 8x + y^{2} = 0$
  2. $x^{2} + 8x + (y - 3)^{2} = 0$
  3. $(x - 3)^{2} + 8x + y^{2} = 0$
  4. $x^{2} + 8x + 8y^{2} = 0$
Reveal answer Fill a bubble to check yourself
B Correct answer
AI explanation

Let the coordinates of M be (X, Y) and the coordinates of B be (a, b). Since point A is (0, 3) and AM = 2AB, we have the section formula relationship X = (2*a + 0)/3 and Y = (2*b + 3)/3, giving a = 3X/2 and b = (3Y - 3)/2. Substituting these values into the given circle equation x^2 + 4x + (y - 3)^2 = 0 yields (9X^2/4) + 6X + 9(Y - 3)^2/4 = 0. Dividing the entire equation by 9/4 results in the locus X^2 + 8X + (Y - 3)^2 = 0.

Keep practicing — related questions