Find the equation of lines through the origin perpendicular to the line 5x+3y+1=0
To find the equation of lines that are perpendicular to the given line, we need to find the slope of the given line and then determine the negative reciprocal of that slope.
Step 1: Rewrite the equation of the given line in slope-intercept form (y = mx + b).
5x + 3y + 1 = 0
3y = -5x - 1
y = (-5/3)x - 1/3
Step 2: Identify the slope of the given line.
The given line's slope is -5/3.
Step 3: Determine the negative reciprocal of the slope.
The negative reciprocal of -5/3 is 3/5.
Step 4: Write the equation of the line in slope-intercept form.
We know that the line passes through the origin (0,0), so we can write the equation as y = mx.
Therefore, the equation of the line through the origin and perpendicular to 5x + 3y + 1 = 0 is y = 3/5x.
A translation takes point (3,4)to(-2,5) find image to the origin under translation
To find the image of the origin under the translation, we can use the fact that a translation is a rigid transformation that preserves distances and orientations. Since we are given the translation of the point (3, 4) to (-2, 5), we can find the translation vector by subtracting the coordinates of the original point from the coordinates of its image:
Translation vector = (x-image - x-original, y-image - y-original) = (-2-3, 5-4) = (-5, 1).
Now, to find the image of the origin, we simply subtract this translation vector from the coordinates of the origin (0, 0):
Image of the origin = (0 - (-5), 0 - 1) = (5, -1).
Therefore, the image of the origin under the given translation is (5, -1).
To find the equation of a line through the origin, we need to determine its slope.
The given equation is 5x + 3y + 1 = 0.
First, we need to rewrite it in slope-intercept form (y = mx + b), where m is the slope and b is the y-intercept.
Rearranging the equation, we get:
3y = -5x - 1.
Dividing both sides by 3, we have:
y = (-5/3)x - 1/3.
The slope of the given line is -5/3.
To find the slope of a line perpendicular to this line, we take the negative reciprocal of -5/3. The negative reciprocal of -5/3 is 3/5.
Therefore, the slope of the line perpendicular to the given line is 3/5.
Now, we can use the slope-intercept form (y = mx) to write the equation of the line through the origin with slope 3/5:
y = (3/5)x.
So, the equation of the line through the origin perpendicular to the line 5x + 3y + 1 = 0 is y = (3/5)x.