in a reflection the image of the line y-2x=3 is the line 2y-x=9.find the axis of reflection.
To find axis of reflection of this question is more complex but l will show how to find easily now let see it
First find the intersection of the two line
y-2x=3)€€€€at x=1 and y=5 ****(1,5)
2y-x=9)
#at this point the axis of reflection passes at(1,5)
*Second take one point from y-2x=3 let (0,3) then find the equation of circle at (1,5) and get radius is the distance between (0,3) and (1,5)=square root of 5
#🤔Then find intersection b/n y-2x=3 and circle (x-1)²+(y-5)²=5......(x-1)²+((2x+3)-5)²=5
=(x-1)²+(2x-2)²=(x-1)²+4(x-1)²..==5(x-1)²=5
(X-1)²=1....x²-2x+1=1******x(x-2)=0
X=0 and x=2........*then (0,3) and (2,7) point on line
y=2x+1.
🤔Then another find intersection between circle and the line 2y-x=9
✍y=(x+9)/2........._(x-1)²+((💧(x+9)/2)-5)²=5 then ....
X= -1 and x =3 then (-1,4) and (3,6) on reflected line 2y=x+9
Now we take our found point that ( smaller x value )on two line that our point (0,3) and (-1,4)
M(0,3)=(-1,4)*********find midpoint
That is (-1/2,7/2) ......then finally we found one point of axis of reflection pass at mid point then use straight line equation between (1,5) and (-1/2,7/2) slope is =1 then
y-5=1(x-1)........y=x-1+5=x+4
Answer is y=x+4...................💯💯💯💯
Today l will show you very simple techniques than previously l posted let me you solution
First find intersection b/n the line and the image reflected line that is (1,5)
🤔find equation of angle bisector to find slope....then form (y-2x-3)/root of 5=+or-(2y-x-9)/root of 5
For +ve ...y-2x-3=2y-x-9**y=-x+6 👈💯💯
For -ve .. y-2x-3=-2y+x+9 ..y=x+4 👈💯💯💯💯💯💯💯💯💯💯
Y=0.988X+4.012 APPROXIMATLY Y=X+4
To find the axis of reflection, we need to find the line that bisects the angle between the original line and its image.
First, let's find the intersection point between these two lines. We can solve the system of equations:
y - 2x = 3
2y - x = 9
By substitution or elimination, we get (1, 5) as the intersection point.
Next, let's find the slope of the original line:
y - 2x = 3
y = 2x + 3
This line has a slope of 2.
Now, let's find the slope of the image line:
2y - x = 9
y = (1/2)x + 9/2
This line has a slope of 1/2.
To find the slope of the angle bisector, we use the formula:
m = (m1 + m2)/2
where m1 and m2 are the slopes of the two lines. Plugging in, we get:
m = (2 + 1/2)/2
m = 5/4
Now we have the slope of the angle bisector, and we also have a point that it passes through (the intersection point (1, 5)). So we can use point-slope form to find the equation of the axis of reflection:
y - 5 = (5/4)(x - 1)
Simplifying, we get:
y = (5/4)x + 15/4
Therefore, the axis of reflection is the line y = (5/4)x + 15/4.
The axis of reflection is the angle bisector of the two lines.
One of the properties of the axis of reflection is that any point lying on the line is equi-distant from both the original line and the image.
Distance of a point from a line Ax+By+C=0 is
±(Ax+By+C)/sqrt(A²+B²)
Can you take it from here?