Find the image of point (8, -5) with respect to the line mirror 5x - 7y = 1.
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A straight line in a two-dimensional plane can be represented by a linear equation. The most common form is the general equation: Ax + By + C = 0. In this video, we'll explore how to work with such equations, particularly in the context of finding the image of a point with respect to a line mirror.
Let's find the image of the point (8, -5) with respect to the line mirror 5x minus 7y equals 1. The process involves three steps. First, we find the equation of the line perpendicular to the mirror that passes through our point. Second, we find where these two lines intersect. Third, we use this intersection point as the midpoint between our original point and its image to calculate the coordinates of the image point.
First, we need to find the equation of the line perpendicular to our mirror that passes through the point (8, negative 5). The mirror line 5x minus 7y equals 1 can be rewritten in slope-intercept form to find its slope, which is 5/7. The slope of a perpendicular line is the negative reciprocal, so negative 7/5. Using the point-slope form with our point (8, negative 5), we get y minus negative 5 equals negative 7/5 times (x minus 8). Simplifying this equation gives us 7x plus 5y equals 31.
Next, we find where the mirror line and the perpendicular line intersect by solving the system of equations. We have 5x minus 7y equals 1 and 7x plus 5y equals 31. To eliminate y, we multiply the first equation by 5 and the second by 7. This gives us 25x minus 35y equals 5 and 49x plus 35y equals 217. Adding these equations eliminates y, resulting in 74x equals 222, so x equals 3. Substituting x equals 3 into the first equation gives us y equals 2. Therefore, the intersection point is (3, 2).
Finally, we use the intersection point (3, 2) as the midpoint between the original point (8, negative 5) and its image to find the coordinates of the image point. Using the midpoint formula, we set (3, 2) equal to ((8 plus x)/2, (negative 5 plus y)/2). Solving for x, we get 3 equals (8 plus x)/2, which gives x equals negative 2. Solving for y, we get 2 equals (negative 5 plus y)/2, which gives y equals 9. Therefore, the image point is (negative 2, 9).
Let's verify our solution. First, we check that the image point (negative 2, 9) lies on the perpendicular line 7x plus 5y equals 31. Substituting the coordinates, we get 7 times negative 2 plus 5 times 9, which equals negative 14 plus 45, or 31. This confirms our point is on the correct line. Second, we verify that (3, 2) is indeed the midpoint of (8, negative 5) and (negative 2, 9) using the midpoint formula. Third, we can confirm that the distance from the original point to the mirror line equals the distance from the image point to the mirror line, which is a fundamental property of reflection.