To find the length of the subtangent, we first need to find the slope of the tangent line at the point (1, y).

To find the slope of the tangent line, we take the derivative of the equation with respect to x:

d/dx (x^2 + xy + y^2) = d/dx (7)

2x + y + x(dy/dx) + 2y(dy/dx) = 0

Simplifying, we get:

dy/dx = -(2x + y)/(x + 2y)

Substituting x = 1, we have:

dy/dx = -(2(1) + y)/(1 + 2y)

To find the value of y, we substitute x = 1 into the equation:

(1)^2 + (1)y + y^2 = 7

1 + y + y^2 = 7

y^2 + y - 6 = 0

Factoring, we get:

(y + 3)(y - 2) = 0

So, y = -3 or y = 2.

Substituting y = -3 into dy/dx = -(2(1) + y)/(1 + 2y), we get:

dy/dx = -(2(1) + (-3))/(1 + 2(-3))

dy/dx = -(2 - 3)/(-5)

dy/dx = 1/5

Substituting y = 2 into dy/dx = -(2(1) + y)/(1 + 2y), we get:

dy/dx = -(2(1) + 2)/(1 + 2(2))

dy/dx = -(2 + 2)/(1 + 4)

dy/dx = -4/5

Therefore, the slope of the tangent line at the point (1, -3) is 1/5, and the slope of the tangent line at the point (1, 2) is -4/5.

The length of the subtangent is given by the formula:

Length = |(y - mx)/sqrt(1 + m^2)|

For the point (1, -3) with slope 1/5:

Length = |(-3 - (1/5)(1))/sqrt(1 + (1/5)^2)|

Length = |(-3 - 1/5)/sqrt(1 + 1/25)|

Length = |-16/5/sqrt(26/25)|

Length = |-16/5/(sqrt(26)/5)|

Length = |-16/sqrt(26)|

For the point (1, 2) with slope -4/5:

Length = |(2 - (-4/5)(1))/sqrt(1 + (-4/5)^2)|

Length = |(2 + 4/5)/sqrt(1 + 16/25)|

Length = |(14/5)/sqrt(41/25)|

Length = |(14/5)/(sqrt(41)/5)|

Length = |14/sqrt(41)|