Find the equation of the chord of contact of the point $(1,2)$ with respect to the circle ${ x }^{ 2 }+{ y }^{ 2 }+2x+3y+1=0$
- $4x+7y+10=0$
- $4x-7y+10=0$
- $7x+4y+10=0$
- $7x-4y+10=0$
The equation of the chord of contact for a circle x^2 + y^2 + 2gx + 2fy + c = 0 from point (x1, y1) is xx1 + yy1 + g(x + x1) + f(y + y1) + c = 0. Substituting (1, 2) and the circle parameters (g=1, f=1.5, c=1): x(1) + y(2) + 1(x + 1) + 1.5(y + 2) + 1 = 0. This simplifies to x + 2y + x + 1 + 1.5y + 3 + 1 = 0, or 2x + 3.5y + 5 = 0. Multiplying by 2 gives 4x + 7y + 10 = 0.
The formula for the chord of contact of tangents drawn from an external point (x1, y1) to a circle x^2 + y^2 + 2gx + 2fy + c = 0 is xx1 + yy1 + g(x + x1) + f(y + y1) + c = 0. Here, the point is (1, 2) and the circle values are g = 1, f = 3/2, and c = 1. Substituting these values gives x(1) + y(2) + 1(x + 1) + (3/2)(y + 2) + 1 = 0, which simplifies to 2x + 7y + 10 = 0 after multiplying by 2. Dividing by 2 yields the final equation 4x + 14y + 20 = 0, which is equivalent to the option form; working through the arithmetic directly gives 4x + 7y + 10 = 0.