1. Key Concepts and Formulas

This problem requires the application of two core concepts from conic sections:

• Equation of a Chord with a Given Midpoint: For a general conic $S \equiv Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$, if $(h, k)$ is the midpoint of a chord, the equation of that chord is given by $T = S_1$.

  • $T$ is obtained by replacing $x^2 \to xh$, $y^2 \to yk$, $xy \to \frac{1}{2}(xk + yh)$, $x \to \frac{1}{2}(x+h)$, $y \to \frac{1}{2}(y+k)$, and the constant term remains unchanged.
  • $S_1$ is obtained by substituting the midpoint coordinates $(h, k)$ into the equation of the conic, i.e., $S_1 = Ah^2 + Bhk + Ck^2 + Dh + Ek + F$.

• Condition for Tangency of a Line to a Parabola: A straight line with equation $y = mx + c$ is tangent to the standard parabola $y^2 = 4ax$ if and only if $c = \frac{a}{m}$.

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2. Step-by-Step Solution

Step 1: Determine the Equation of the Chord of the Hyperbola

We are given the hyperbola $x^2 - y^2 = 4$. We can rewrite this as $S \equiv x^2 - y^2 - 4 = 0$. Let $(h, k)$ be the midpoint of a chord of this hyperbola. Our goal is to find the locus of $(h, k)$.

The equation of a chord whose midpoint is $(h, k)$ is given by the formula $T = S_1$.

• Calculate $T$ (Tangent at $(h,k)$ form): To find $T$, we apply the substitution rules to the hyperbola equation $x^2 - y^2 - 4 = 0$. Replace $x^2$ with $xh$ and $y^2$ with $yk$. The constant term remains unchanged. $T = xh - yk - 4$

• Calculate $S_1$ (Value of S at $(h,k)$): To find $S_1$, we substitute the coordinates of the midpoint $(h, k)$ into the equation of the hyperbola. $S_1 = h^2 - k^2 - 4$

• Equate $T = S_1$ to get the Chord Equation: Now, we set $T$ equal to $S_1$: $xh - yk - 4 = h^2 - k^2 - 4$ The constant term $-4$ cancels out from both sides: $xh - yk = h^2 - k^2$

  Explanation: The $T=S_1$ formula is a powerful shortcut. It essentially represents the equation of a line that passes through the midpoint $(h,k)$ and is perpendicular to the diameter passing through $(h,k)$. This is precisely the definition of a chord with a given midpoint.

  To prepare for the tangency condition, we need to express this chord equation in the slope-intercept form $y = mx + c$. Rearranging the equation: $yk = xh - (h^2 - k^2)$ Assuming $k \neq 0$ (if $k=0$, the chord is vertical, which cannot be tangent to $y^2=8x$ unless $h=0$ which is a degenerate case), we can divide by $k$: $y = \left(\frac{h}{k}\right)x - \left(\frac{h^2 - k^2}{k}\right)$ By comparing this with $y = mx + c$, we identify the slope $m$ and y-intercept $c$ of the chord: $m = \frac{h}{k} \quad \text{and} \quad c = -\frac{h^2 - k^2}{k}$

Step 2: Apply the Tangency Condition to the Parabola

The problem states that these chords touch the parabola $y^2 = 8x$. To use the tangency condition $c = a/m$, we first need to identify the parameter $a$ for the given parabola. We compare $y^2 = 8x$ with the standard form $y^2 = 4ax$. From this comparison, we see that $4a = 8$, which implies $a = 2$.

Now, we substitute the values of $m$, $c$ (from Step 1), and $a$ into the tangency condition $c = a/m$: $-\frac{h^2 - k^2}{k} = \frac{2}{h/k}$

Step 3: Simplify and Determine the Locus

Now, we simplify the equation obtained in Step 2: $-\frac{h^2 - k^2}{k} = \frac{2k}{h}$ To eliminate the denominators, we multiply both sides by $hk$ (assuming $h \neq 0$ and $k \neq 0$): $-h(h^2 - k^2) = 2k^2$ Distribute the $-h$ on the left side: $-h^3 + hk^2 = 2k^2$ We want to find the locus, which means an equation relating $h$ and $k$. Let's rearrange the terms to isolate $k^2$: $hk^2 - 2k^2 = h^3$ Factor out $k^2$ from the terms on the left side: $k^2(h - 2) = h^3$

Finally, to express the locus, we replace the general midpoint coordinates $(h, k)$ with the standard variables $(x, y)$: $y^2(x - 2) = x^3$

This equation represents the locus of the midpoints of the chords.