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Question 392130: Hello tutor,
This one question on my online homework assignment has been puzzling me for a while.
A circle is given centered at (0,0)with a radius of 2.
A line cuts through the center of the circle (slope of line unknown) but the line makes an angle of pi/3 with the x axis in the fourth quadrant.
This line continues and cuts through point Q (coordinates unknown) of the circle in the fourth quadrant.
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Another line forms a tangent to the circle at point Q and is perpendicular to the original line.
this second line ( which is tangent to the circle and perpendicular to the original line) continues and crosses the x axis at some point P (coordinates unknown)
* note that both lines share a point Q which lies on the circle.
We are asked to ultimately find the coordinates of point P (x , 0)
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there is an order of steps we have to perform.
a)Find the coordinates of the point Q (x,y)
b)Knowing two points on the original line, namely (0,0) and Q,compute the slope of the dotted line
c)Knowing the slope of the first line, Compute the slope of the second line (which is perpendicular to the first line)
d)We now know the point Q on the second line and the slope of the second line, so we find the equation of the line in the form
y=mx+b
e) therefor the coordinates of point P are (x,0) (Find x)
Answer by lwsshak3(11628)   (Show Source):
You can put this solution on YOUR website! A circle is given centered at (0,0)with a radius of 2.
A line cuts through the center of the circle (slope of line unknown) but the line makes an angle of pi/3 with the x axis in the fourth quadrant.
This line continues and cuts through point Q (coordinates unknown) of the circle in the fourth quadrant.
------------------------------------------------------------------------------
Another line forms a tangent to the circle at point Q and is perpendicular to the original line.
this second line ( which is tangent to the circle and perpendicular to the original line) continues and crosses the x axis at some point P (coordinates unknown)
* note that both lines share a point Q which lies on the circle.
We are asked to ultimately find the coordinates of point P (x , 0)
..
According to the information given, I ended with a right triangle in quadrant IV, labeled as follows"
O - center of circle
Q - point on circle where second line is tangent to circle and perpendicular to first line which goes thru origin.
P - point at which second line crosses x-axis
Angle at O - 60 deg
Angle at P - 30 deg
Angle at Q - 90 deg
Line segment OQ is equal to the radius = 2
since the reference angle of the first line is pi/3 = 60 deg, the angle at P must be 30 deg. This makes OP, the hypotenuse of the triangle twice that of OQ, being that OQ is opposite a 30 degree angle. Therefore the hypotenuse is = 4.
..
ans: The coordinates at point P is (4,0)
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