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Calculus/Points on lines and circles

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Expert: - 1/9/2011


Question

problem
This one question on my online homework assignment has been puzzling me for a while.
I have attached an image of the diagram given for the question.
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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
Hi Shahzeb,
a) The length of the dotted line from the origin to the point Q(x,y) is 2, the radius of the circle. Using the basic trigonometric formula with the right triangle that includes the angle π/3, then
cos(π/3) = x/2
1/2 = x/2
x = 1
and
sin(π/3) = -y/2  (the minus sign only showing that y would be negative in the fourth quadrant)
√3/2 = -y/2
y = -√3
So, Q has coordinates (1,-√3)

b) The (dotted) line joining points (0,0) and (1,√3) has a slope
m = -√3-0 / 1-0
= -√3

Note that we could simply have found the slope of this line by using the formula m = tanθ where θ is the angle the line makes with the positive x-axis (going anti-clockwise). Here, θ would be π - π/3 = 2π/3 and m = tan(2π/3) = -√3

c) The product of the slopes of two perpendicular lines is equal to -1. So, the slope of the tangent is -1/-√3 = 1/√3 = √3/3

d) The equation of the tangent, with one point and the slope known can be gotten from
y - -√3 / x - 1 = √3/3
y+√3/x-1 = √3/3
3y + 3√3 = x√3 - √3
3y = x√3 - 4√3
y = x√3/3 - 4√3/3

e) P is the point where y = 0 and so
0 = x√3/3 - 4√3/3
x√3/3 = 4√3/3
x = 4
The coordinates of P is therefore (4,0)

Regards  

Ahmed Salami

Expertise

I can provide good answers to questions dealing in almost all of mathematics especially from A`Level downwards. I believe i would be very helpful in calculus and can as well help a good deal in Physics with most emphasis directed towards mechanics.

Experience

An engineering graduate. I have been doing maths and physics all my life.

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