Subtangent to the curve

The subtangent to a curve at a given point is the line segment that touches the curve at that point and is parallel to the tangent at that point.

Given curve and point

The given curve is x^2 + xy + y^2 = 7. Let's find the equation of the tangent at point (1,-3).

Differentiating the curve with respect to x, we get:

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

At point (1,-3), we have x = 1 and y = -3. Substituting these values, we get:

2(1) + (-3) + 1(dy/dx) + 2(-3)(dy/dx) = 0

Simplifying, we get:

-5(dy/dx) = 4

dy/dx = -4/5

So the equation of the tangent at (1,-3) is:

y + 3 = (-4/5)(x - 1)

Simplifying, we get:

4x + 5y = -8

Length of subtangent

The subtangent is parallel to the tangent and passes through (1,-3). So its equation is:

4x + 5y = k (where k is a constant)

Substituting (1,-3), we get:

4(1) + 5(-3) = k

k = -17

So the equation of the subtangent is:

4x + 5y = -17

To find its length, we need to find the distance between (1,-3) and the point where the subtangent intersects the x-axis.

When y = 0, we have:

4x = 17

x = 17/4

So the point of intersection is (17/4,0).

Using the distance formula, we get:

distance = sqrt((17/4 - 1)^2 + (0 - (-3))^2)

Simplifying, we get:

distance = sqrt(225)/4 = 15/2

Therefore, the length of the subtangent is 15. Hence, option C is the correct answer.