Key Concepts and Formulas

• Equation of a Circle: $(x-h)^2 + (y-k)^2 = r^2$, where $(h, k)$ is the center and $r$ is the radius.
• General Equation of a Circle: $x^2 + y^2 + 2gx + 2fy + c = 0$, where the center is $(-g, -f)$ and the radius is $\sqrt{g^2 + f^2 - c}$.
• Polar of a Point: The equation of the polar of a point $(x_1, y_1)$ with respect to the circle $x^2 + y^2 + 2gx + 2fy + c = 0$ is given by $xx_1 + yy_1 + g(x + x_1) + f(y + y_1) + c = 0$.

Step-by-Step Solution

Step 1: Find the General Equation of the Circle

The given circle has its center at $(2, 3)$ and a radius of $4$. Therefore, its equation is: $(x - 2)^2 + (y - 3)^2 = 4^2$ $(x - 2)^2 + (y - 3)^2 = 16$ Expanding the equation to get the general form: $x^2 - 4x + 4 + y^2 - 6y + 9 = 16$ $x^2 + y^2 - 4x - 6y + 13 - 16 = 0$ $x^2 + y^2 - 4x - 6y - 3 = 0$

Step 2: Determine the Coefficients for the Polar Equation

Comparing the equation $x^2 + y^2 - 4x - 6y - 3 = 0$ with the general form $x^2 + y^2 + 2gx + 2fy + c = 0$, we have: $2g = -4 \Rightarrow g = -2$ $2f = -6 \Rightarrow f = -3$ $c = -3$

Step 3: Formulate the Equation of the Polar of S(α, β)

Let $S$ be the point $(\alpha, \beta)$. Using the polar formula $xx_1 + yy_1 + g(x + x_1) + f(y + y_1) + c = 0$, with $x_1 = \alpha$, $y_1 = \beta$, $g = -2$, $f = -3$, and $c = -3$: $x\alpha + y\beta - 2(x + \alpha) - 3(y + \beta) - 3 = 0$ Expanding and collecting terms: $x\alpha + y\beta - 2x - 2\alpha - 3y - 3\beta - 3 = 0$ $(\alpha - 2)x + (\beta - 3)y - 2\alpha - 3\beta - 3 = 0$

Step 4: Compare the Polar Equation with the Given Line PQ

The given line PQ is $x + y = 3$, which can be rewritten as $x + y - 3 = 0$. Since the equation derived in Step 3 and the given line represent the same line, their coefficients must be proportional: $\frac{\alpha - 2}{1} = \frac{\beta - 3}{1} = \frac{-2\alpha - 3\beta - 3}{-3}$ From the first two ratios, we have: $\alpha - 2 = \beta - 3$ $\alpha = \beta - 1 \quad (Equation \ 1)$ From the first and third ratios, we have: $\frac{\alpha - 2}{1} = \frac{2\alpha + 3\beta + 3}{3}$ $3(\alpha - 2) = 2\alpha + 3\beta + 3$ $3\alpha - 6 = 2\alpha + 3\beta + 3$ $\alpha - 3\beta = 9 \quad (Equation \ 2)$

Step 5: Solve for α and β

Substitute Equation 1 into Equation 2: $(\beta - 1) - 3\beta = 9$ $-2\beta - 1 = 9$ $-2\beta = 10$ $\beta = -5$ Now, substitute $\beta = -5$ into Equation 1: $\alpha = -5 - 1$ $\alpha = -6$ So, $S = (\alpha, \beta) = (-6, -5)$.

Step 6: Evaluate 4α - 7β

$4\alpha - 7\beta = 4(-6) - 7(-5)$ $= -24 + 35$ $= 11$

Common Mistakes & Tips

• Sign Errors: Be extra cautious with signs when expanding and rearranging equations. A single sign error can lead to a wrong answer.
• Proportionality: When comparing coefficients of proportional lines, ensure that the constant terms also maintain the same ratio.
• General vs. Standard Form: Remember to convert the equation to the correct general form before applying the polar formula.

Summary

We found the equation of the polar of the point S$(\alpha, \beta)$ with respect to the given circle. Then, we compared the coefficients of this equation with the given line $x+y=3$ to establish a relationship between $\alpha$ and $\beta$. Solving the resulting equations, we determined the values of $\alpha$ and $\beta$ and finally calculated the value of the expression $4\alpha - 7\beta$, which is 11.

Final Answer

The final answer is \boxed{11}.