Question:

Step 1: Set up the problem using the mid-point chord relation. Let $M(x, y)$ be the mid-point of a chord of the circle $x^2 + y^2 = r^2$ that passes through the external point $(p, q)$. For a chord of a circle with mid-point $(x, y)$, the equation of the chord is given by the relation $T = S_1$, where: $$xx_1 + yy_1 = x_1^2 + y_1^2$$ This gives us the chord equation: $$xX + yY = x^2 + y^2$$ where $(X, Y)$ are coordinates of points on the chord. Step 2: Apply the condition that the chord passes through $(p, q)$. Since the chord passes through the point $(p, q)$, we substitute $X = p$ and $Y = q$ into the chord equation: $$xp + yq = x^2 + y^2$$ Step 3: Rearrange to obtain the locus equation. Rearranging the equation from Step 2: $$x^2 + y^2 - xp - yq = 0$$ or equivalently: $$x^2 + y^2 - px - qy = 0$$ This is the locus of the mid-point $(x, y)$ of all chords passing through $(p, q)$. Step 4: Compare coefficients with the given form. The locus equation is $x^2 + y^2 - px - qy = 0$. We compare this with the given form $x^2 + 2hxy + y^2 + 2gx + 2fy + c = 0$: - Coefficient of $xy$: $0 = 2h$ $\Rightarrow$ $h = 0$ - Coefficient of $x$: $-p = 2g$ $\Rightarrow$ $g = -\frac{p}{2}$ - Coefficient of $y$: $-q = 2f$ $\Rightarrow$ $f = -\frac{q}{2}$ - Constant term: $c = 0$ Step 5: Verify against the given options. Checking each option: - Option 1: $h = pq$ — We found $h = 0$, so this is **incorrect**. - Option 2: $g = p$ — We found $g = -\frac{p}{2}$, so this is **incorrect**. - Option 3: $f = q$ — We found $f = -\frac{q}{2}$, so this is **incorrect**. - Option 4: $c = 0$ — We found $c = 0$, so this is **correct**. **Final Answer: Option 4** ($c = 0$) is the correct answer.