Problem Breakdown:

We need to determine the product of the slope and $y$-intercept of the line:

1. Perpendicular to BC (line passing through points $B(4,1)$ and $C(3,-5)$),
2. Passing through point $A(-3,-1)$.

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Step 1: Calculate the slope of line BC

The slope formula between two points $(x_1, y_1)$ and $(x_2, y_2)$ is:

$m = \frac{y_2 - y_1}{x_2 - x_1}$

For $B(4,1)$ and $C(3,-5)$: $m_{BC} = \frac{-5 - 1}{3 - 4} = \frac{-6}{-1} = 6$

So, the slope of $BC$ is $6$.

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Step 2: Slope of the perpendicular line

The slope of a line perpendicular to another is the negative reciprocal. Therefore:

$m_{\perp} = -\frac{1}{m_{BC}} = -\frac{1}{6}$

The slope of the line perpendicular to $BC$ is $-\frac{1}{6}$.

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Step 3: Equation of the perpendicular line passing through A

The equation of a line is given by:

$y - y_1 = m(x - x_1)$

Here, $m = -\frac{1}{6}$, and the point is $A(-3, -1)$. Substituting:

$y - (-1) = -\frac{1}{6}(x - (-3))$ $y + 1 = -\frac{1}{6}(x + 3)$ $y + 1 = -\frac{1}{6}x - \frac{1}{2}$ $y = -\frac{1}{6}x - \frac{1}{2} - 1$ $y = -\frac{1}{6}x - \frac{3}{2}$

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Step 4: Slope and $y$-intercept

The slope ($m$) of this line is $-\frac{1}{6}$, and the $y$-intercept ($c$) is $-\frac{3}{2}$.

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Step 5: Product of slope and $y$-intercept

The product of the slope and $y$-intercept is:

$m \cdot c = \left(-\frac{1}{6}\right) \cdot \left(-\frac{3}{2}\right)$ $m \cdot c = \frac{1}{6} \cdot \frac{3}{2} = \frac{3}{12} = \frac{1}{4}$

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Final Answer:

The product of the slope and $y$-intercept is:

$\boxed{\frac{1}{4}}$

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Would you like me to elaborate on any step or explore related problems?

Related Questions:

1. How do you find the equation of a line perpendicular to a given line?
2. What is the significance of the negative reciprocal for perpendicular slopes?
3. How do you find the product of slope and $y$-intercept geometrically?
4. How would the answer change if the perpendicular line passed through a different point?
5. What is the role of the slope-intercept form in solving such problems?

Tip:

Always confirm the relationship between slopes of perpendicular lines by calculating the product of their slopes ($-1$).