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Question 515164: 1)Determine the equation(s) the circle (s) touching the lines x+1=0,x-9=0, and y-4=0.
2) Determine the coordinates of the points where the line x+2y-7=0 cuts the circle, and sketch a diagram to show the region for which both x^2+y^2-4x<21 and x+2y>7.
Answer by lwsshak3(11628)   (Show Source):
You can put this solution on YOUR website! 1)Determine the equation(s) the circle (s) touching the lines x+1=0,x-9=0, and y-4=0.
2) Determine the coordinates of the points where the line x+2y-7=0 cuts the circle, and sketch a diagram to show the region for which both x^2+y^2-4x<21 and x+2y>7.
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1) Circle is tangent to x=-1, x=9, and y=4. The perpendicular distance between x=-1 and x=9 is 10 units which is the diameter of the circle and makes the x-coordinate of the center=4.
The y-coordinate of the center must be 5 units down from y=4 to y=-1.
Equation of circle: (x-4)^2+(y+1)^2=25
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Where line cuts circle:
2) x^2+y^2-4x<21
complete the square
(x-2)^2+y^2=25
y^2=(25-(x-2)^2)
take sqrt of both sides
y=(25-(x-2)^2)^.5
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x+2y-7=0
2y=-x+7
y=-x/2+7/2=(-x+7)/2
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finding points of intersection for line and circle
equate line and circle equations
(25-(x-2)^2)^0.5=(-x+7)/2
square both sides
25-(x-2)^2=(x^2-14x+49)/4
25-x^2+4x-4=(x^2-14x+49)/4
100-4x^2+16x-16=x^2-14x+49
5x^2+30x-35=0
x^2-6x-35=0
(x+1)(x-7)=0
x=-1 and 7
y=(-x+7)/2=(1+7)/2=4
y=(-7+7)2=0
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line cuts circle at (-1,4) and (7,0)
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Show region for which both x^2+y^2-4x<21 and x+2y>7.
x^2+y^2-4x<21
x^2+y^2-4x=21
complete the square
(x^2-4x+4)+y^2=21+4
(x-2)^2+y^2=25
This is a circle whose radius<25
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finding points of intersection for line and circle
y^2=25-(x-2)^2
y=(25-(x-2)^2)^0.5 (circle)
y=(-x+7)/2 (line)
(25-(x-2)^2)^0.5=(-x+7)/2
square both sides
25-(x-2)^2=(-x+7)^2/4
25-x^2+4x-4=(x^2-14x+49)/4
100-4x^2+16x-16=x^2-14x+49
5x^2-30x-35=0
x^2-6x-7=0
(x+1)(x-7)=0
x=-1 and 7
y=(-x+7)/2=(-1+7)/2=8/2=4
y=(-7+7)/2=0
Line cuts circle at (-1,4) and (7,0)
The region, therefore, is the segment of the circle bounded by an arc of the circle and the straight line (chord) that cuts the circle at (-1,4) and (7,0)
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