Question

I need help in Analytical Geometry
Find the equation of a circle tangent to the lines 2x+y=4, 2x+y=2 and x-2y+5=0

Answer 1

@Nnesha can you help my friend?

Answer 2

@nincompoop

Answer 3

@Lord2clash @ConnexusStudent0912 @chainedecho can someone help with the geometry?

Answer 4

Have you done any similar problems to this one ? I can guess at an approach, but it seems complicated.

Answer 5

not yet. our professor gave this as our assignment

Answer 6

I would plot the 3 lines, and they should make a triangle.
Now the problem is finding the "in-circle" of a triangle.

Answer 7

the center of the circle is where the "angle bisectors" (of the triangle) meet.

Answer 8

though looking at the equations
2x+y=4,
2x+y=2
x-2y+5=0

the first 2 are parallel, so the distance between those two lines is the diameter of the circle

Answer 9

@phi is my figure right?

Answer 10

is this correct?

Answer 11

yes, here is a geogebra plot (geogebra is a free download)

Answer 12

the center of the circle will be on the line that bisects the two parallel lines
it will be a distance of 1 radius from the perpendicular line.
because we can go either "up" or "down", there are two different points that could be the center of the circle.

Answer 13

can you help me with the solution to find the general equation of that problem?

Answer 14

The first step is to find the distance between the two parallel lines
any idea how to do that ? we could use vectors if you have learned that approach.
or, if you look at your 3 lines in slope-intercept form:
y= -2x +4
y= -2x +2
y= (½) x + 5/2

you see the 3rd line has a slope of ½ which is the negative reciprocal of -2 (the slope of the two other lines)
so we could find the intersection points and then find the distance between two points using the distance formula.

Answer 15

* the 3rd line is perpendicular to the other two lines

Answer 16

\[\sqrt{(x _{2}-x _{1)}+(y _{2}-y _{1)}}\]

Answer 17

is this the distance formula that you're talking about or the one with r^2?

Answer 18

yes, that is the distance formula
but first we need the two points

Answer 19

you can find point A by solving
2x+y=4
x-2y= -5

multiply the 2nd equation by -2 (all terms, both sides) and you get
-2x+4y= 10

add the two equations term by term
2x+y=4
-2x+4y= 10
-----------
0 + 5y= 14
y=14/5 or 2.8
and using the 2nd equation, x -2*2.8= -5
x= -5+5.6 = 0.6

so point A is at (0.6, 2.8)