Distance Between Points of Intersection of Mirror Image Curve

To find the distance between the points of intersection $A$ and $B$ of the curve $C$ with the line $y = -5$, we first determine the equation of the curve $C$.

The curve $C$ is the reflection of the parabola $y^2 = 4x$ with respect to the straight line $x + y + 4 = 0$.

Let $(x', y')$ be any point on the curve $C$, which is the reflection of a point $(x, y)$ on the parabola $y^2 = 4x$ across the line $x + y + 4 = 0$. Using the formula for the reflection of a point $(x', y')$ across a line $ax + by + c = 0$:

$\frac{x - x'}{a} = \frac{y - y'}{b} = -2 \frac{a x' + b y' + c}{a^2 + b^2}$

Substituting $a = 1$, $b = 1$, and $c = 4$:

$\frac{x - x'}{1} = \frac{y - y'}{1} = -2 \frac{x' + y' + 4}{1^2 + 1^2} = -(x' + y' + 4)$

Solving for $x$ and $y$ in terms of $x'$ and $y'$: $x = x' - (x' + y' + 4) = -y' - 4$ $y = y' - (x' + y' + 4) = -x' - 4$

Since the point $(x, y)$ lies on the parabola $y^2 = 4x$, we substitute $x = -y' - 4$ and $y = -x' - 4$ into its equation:

$(-x' - 4)^2 = 4(-y' - 4)$ $(x' + 4)^2 = -4(y' + 4)$

Thus, the equation of the reflected curve $C$ is: $(x + 4)^2 = -4(y + 4)$

To find the points $A$ and $B$, we set $y = -5$ in the equation of curve $C$:

$(x + 4)^2 = -4(-5 + 4)$ $(x + 4)^2 = 4$ $x + 4 = \pm 2$

This gives two values for $x$: $x = -4 + 2 = -2$ $x = -4 - 2 = -6$

Therefore, the points of intersection are $A(-2, -5)$ and $B(-6, -5)$.

The distance between $A$ and $B$ along the horizontal line $y = -5$ is:

$AB = |-2 - (-6)| = 4$