Multiple choice

The chord of contact of $(2,1)$ w.r.t to the circle ${ x }^{ 2 }+{ y }^{ 2 }+4x+4y+1=0$ is

  1. $2x+y-7=0$
  2. $4x+3y+7=0$
  3. $3x+4y+1=0$

  4. not existing
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The chord of contact for a point (x1, y1) with respect to x^2 + y^2 + 2gx + 2fy + c = 0 is xx1 + yy1 + g(x + x1) + f(y + y1) + c = 0. Substituting (2, 1) and the circle equation: 2x + 1y + 2(x + 2) + 2(y + 1) + 1 = 0, which simplifies to 4x + 3y + 7 = 0.

AI explanation

The equation of the chord of contact for a point (x1, y1) with respect to a circle is found using the formula T = 0, which means replacing x^2 with xx1, y^2 with yy1, x with (x + x1)/2, and y with (y + y1)/2. Substituting the point (2, 1) into the circle equation x^2 + y^2 + 4x + 4y + 1 = 0 yields 2x + y + 4((x + 2)/2) + 4((y + 1)/2) + 1 = 0. Simplifying this gives 4x + 3y + 7 = 0.

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