Let's analyze the problem presented in the image and solve it.

Problem Recap:

• You are given two points $A(-2, 3)$ and $B(4, -2)$.
• The question asks for the equation of the straight line passing through the point $P(1, 1)$ that is perpendicular to the line $AB$.

Solution:

1. Find the Slope of Line $AB$:

   • The slope of a line through two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by: $m_{AB} = \frac{y_2 - y_1}{x_2 - x_1}$
   • For points $A(-2, 3)$ and $B(4, -2)$, the slope $m_{AB}$ is: $m_{AB} = \frac{-2 - 3}{4 - (-2)} = \frac{-5}{6}$

2. Find the Slope of the Perpendicular Line:

   • The slope of a line perpendicular to another is the negative reciprocal of the original line's slope.
   • If $m$ is the slope of a line, the slope of a line perpendicular to it is $-\frac{1}{m}$.
   • Therefore, the slope of the line perpendicular to $AB$ is: $m_{\perp} = -\frac{1}{-\frac{5}{6}} = \frac{6}{5}$

3. Write the Equation of the Line through $P(1, 1)$ with Slope $\frac{6}{5}$:

   • The equation of a line with slope $m$ passing through a point $(x_1, y_1)$ is given by: $y - y_1 = m(x - x_1)$
   • Substituting $m = \frac{6}{5}$, $x_1 = 1$, and $y_1 = 1$, we get: $y - 1 = \frac{6}{5}(x - 1)$
   • Simplifying the equation: $y - 1 = \frac{6}{5}x - \frac{6}{5}$ $y = \frac{6}{5}x - \frac{6}{5} + 1$ $y = \frac{6}{5}x - \frac{1}{5}$

Thus, the equation of the line passing through $P(1, 1)$ and perpendicular to $AB$ is: $y = \frac{6}{5}x - \frac{1}{5}$

Would you like further explanation or have any other questions?

Related Questions:

1. How do you find the equation of a line parallel to a given line passing through a specific point?
2. Can you determine the point of intersection between the line you just found and the line $AB$?
3. What is the distance between point $P(1, 1)$ and the line $AB$?
4. How would the equation change if the line was parallel to $AB$ instead of perpendicular?
5. How do you find the equation of a line in general form given two points?

Tip:

Always double-check the signs when calculating slopes and remember that the perpendicular slope is the negative reciprocal. This is a common area where errors can occur.