What is the equation of a line through (-2,5) and whose segment intercepted between the axes in the 2nd quadrant is 7sqrt2?
Please help me solve this
Where are you getting these ?
let P be (-2,5)
let the x-intercept be A(x,0) and the y-intercept be B(0,y)
so first of all:
x^2 + y^2 = (7√2)^2 = 98
now the slope of AP = slop BP
(y-5)/(2 = -5/(x+2)
cross-multiplying and re-arranging I got
xy + 2y = 5x
y(x+2) = 5x
y = 5x/(x+2)
sub that into above:
x^2 + y^2 = 98
x^2 + 25x^2/(x+2)^2 = 98
x^2(x+2)^2 + 25x^2 = 98(x+2)^2
expanding all this and simplifying I got the horrible equation
x^4 + 4x^3 - 69x^2 - 392x - 392 = 0
at this point I "cheated" and ran it through Wolfgram
http://www.wolframalpha.com/input/?i=x%5E4+%2B+4x%5E3+-+69x%5E2+-+392x+-+392+%3D+0
to get an exact value of x = -7
so y = -35/(-5) = 7
we can now find the slope of our line which is
(5-0)/(-2+7) = 1, wow!
equation is
y = x + 7
To find the equation of a line through a given point and with a certain segment intercepted between the axes, we can use the point-slope form of the equation of a line.
Step 1: Identify the given information:
- Point (-2, 5) on the line
- Segment intercepted between the axes in the 2nd quadrant is 7√2
Step 2: Determine the slope of the line:
Since the segment intercepted between the axes in the 2nd quadrant is positive and along the y-axis, the slope of the line is negative. Let's denote it as m.
Step 3: Calculate the slope:
The length of the segment intercepted between the axes can be calculated using the distance formula:
Length = √((-2 - 0)^2 + (5 - 0)^2) = √((-2)^2 + 5^2) = √(4 + 25) = √29
Since the length of the segment is 7√2, we can set up the following equation:
√29 = m * (7√2)
Solve for m:
√29 = 7√2 * m
m = √29 / (7√2)
m = (√29 * √2) / (7 * 2)
m = (√(29*2)) / (7 * 2)
m = (√58) / 14
Step 4: Write the equation of the line using point-slope form:
The point-slope form of the equation is:
y - y1 = m(x - x1)
Substituting the values, we have:
y - 5 = (√58 / 14)(x + 2)
This is the equation of the line passing through (-2, 5) with the segment intercepted between the axes in the 2nd quadrant being 7√2.