question from cordinates and straight lines?

Question

1)The equation of the straight line which bisects the intercepts made by the axes on the lines x+y=2 and 2x+3y=6 is ?

2)the equation of a straight line passing through the point (-2,3)and making intercepts of equal lenght on the axes is ?

Answer 1

1) x+y=2 has intercepts (0,2),(2,0).
The mid point of the segment joining these intercepts is
[(0+2)/2,(2+0)/2]=(1,1).
The perpendicular line will have a slope of -1/slope of x+y=2.
or y=2-x so this has a slope of -1, and thus the perpendicular has a slope of -1/-1=1.

So it is a line with a slope of 1 and passing through (1,1)
y=1+1(x-1) or
y=x
------------------------------------------------------------------------------
2x+3y = 6 has intercepts (0,2),(3,0).
The mid point of the segment joining these intercepts is
[(0+2)/2,(3+0)/2]=(1,3/2).
The perpendicular line will have a slope of -1/slope of 2x+3y=6.
or y=2-2x/3 so this has a slope of -2/3, and thus the perpendicular has a slope of -1/(-2/3) = 3/2..

So it is a line with a slope of 3/2 and passing through (1,3/2)
y=3/2+3/2(x-1) or
y= 3/2x

==================
2) If it makes equal intercepts, it has the form y=a-x or y = a+x where a is the length of the intercept and m the slope.

so 3= a-(-2) or a = 1, giving y = 1-x
or 3=a+(-2) or a = 5, giving y=5+x

These two equations are the ones required.

Answer 2

1) First find the intercepts and midpoint of both lines.

The intercepts of x + y = 2 are (2,0) and (0,2).
The midpoint is (1,1).

The intercepts of 2x + 3y = 6 are (3,0) and (0,2).
The midpoint is (3/2,1).

Now find the equation of a line passing thru both of the midoints.

The slope m = Δy/Δx = (1 - 1)/(3/2 - 1) = 0
So the line is horizontal. y is a constant.

The equation of the line is y = 1.

2) The line passes thru P(-2,3) and thru (a,0) and (0,a).

First find the slope of the line.
The slope m = Δy/Δx = (0 - a)/(a - 0) = -a/a = -1

The equation of the line is y = mx + b

Plugging in values we have:

3 = -1(-2) + b
3 = 2 + b
b = 1

So the equation of the line is:

y = -x + 1

Answer 3

/a million/ the line //-el to x+y=a million thro pt A=[a million,a million] looks x+y=2; 2 lines x+y=2 and 2x-3y= 4 intersect in pt B=[2,0]; the area d = |a million,a million] looks x+y=2; 2 lines x+y=2 and 2x-3y= 4 intersect in pt B=[2,0]; the area d = |B-A| = sqrt((2-a million)^2 +(0-a million)^2) = sqrt(2); 2 (draw a photo!) the quadrilateral is rectangle with corners O=[0,0], A=[2,0], B=[2,3], C=[0,3] and area s= 2*3 =6; now the line L y = 0.25*x + b is //-el to the given y = 0.25x, b must be stumbled on; as L divides the rectangle into 2 equivalent aspects then it cuts the vertical aspects of rectangle D=[0,b] and E=[2, 3-b| = sqrt((2-a million)^2 +(0-a million)^2) = sqrt(2); 2 (draw a photo!) the quadrilateral is rectangle with corners O=[0,0], A=[2,0], B=[2,3], C=[0,3] and area s= 2*3 =6; now the line L y = 0.25*x + b is //-el to the given y = 0.25x, b must be stumbled on; as L divides the rectangle into 2 equivalent aspects then it cuts the vertical aspects of rectangle D=[0,b] and E=[2, 3-b], so plug pt E into L: 3-b = 0.25*2 +b, thence b=a million.25;