What are the equations of the lines through (-5,-3) and passing at distance 2sqrt5 from (5,7)
Please I really need your help. I do not know how to do this problem.
you are looking for two of the radii of the circle (x+5)^2 + (y+3)^2 = r^2
r^2 equals the square of the distance from (-5,-3) to (5,7), minus (2√5)^2
the radii are tangent to
(x-5)^2 + (y-7)^2 = 20
the points of tangency (intersections of the two circles) will give the points needed (along with (-5,-3))to write the equations of the lines
To find the equations of the lines passing through (-5,-3) and at a distance of 2√5 from (5,7), we can follow these steps:
Step 1: Determine the midpoint between (-5,-3) and (5,7).
To find the midpoint, we use the midpoint formula:
Midpoint = ((x1 + x2)/2, (y1 + y2)/2)
Substituting the values, we get:
Midpoint = ((-5 + 5)/2, (-3 + 7)/2)
Midpoint = (0, 2)
Step 2: Find the slope of the line passing through (-5,-3) and (5,7).
The slope can be found using the slope formula:
Slope (m) = (y2 - y1) / (x2 - x1)
Substituting the values, we get:
Slope (m) = (7 - (-3)) / (5 - (-5))
Slope (m) = 10 / 10
Slope (m) = 1
Step 3: Find the equations of the lines passing through (0,2) with a slope of 1 and a distance of 2√5 from (5,7).
There will be two possible lines, one on each side of the given point at a distance of 2√5.
To find the equations of the lines, we can make use of the point-slope form of a line, which is:
y - y1 = m(x - x1)
For the line on one side of (0,2) passing through (0,2):
Using the point-slope form, we substitute the values as follows:
y - 2 = 1(x - 0)
y - 2 = x
For the line on the other side of (0,2) passing through (0,2):
Similarly, we substitute the values into the point-slope form:
y - 2 = -1(x - 0)
y - 2 = -x
So, the equations of the lines passing through (-5,-3) and at a distance of 2√5 from (5,7) are:
y - 2 = x and y - 2 = -x