Problem 4.2
We're told that y = 2x is the equation of the tangent line through point Q.
Plug this into the equation of the circle (that was given at the very top of the page). Solve for x.
x^2 + y^2 - 18x - 6y + 45 = 0
x^2 + (2x)^2 - 18x - 6(2x) + 45 = 0 ... every "y" replaced with "2x"
x^2 + 4x^2 - 18x - 12x + 45 = 0
5x^2 - 30x + 45 = 0
5(x^2 - 6x + 9) = 0
5(x-3)^2 = 0
(x-3)^2 = 0
x-3 = 0
x = 3
This then means y = 2x = 2*3 = 6
The coordinates of point Q are (x,y) = (3,6)
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<h3>
Answer: (3,6)</h3>
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Problem 4.3
We first need to find the location of point S. That will involve solving this system of equations
![\begin{cases}x-2y+12=0\\y = 2x\end{cases}](https://tex.z-dn.net/?f=%5Cbegin%7Bcases%7Dx-2y%2B12%3D0%5C%5Cy%20%3D%202x%5Cend%7Bcases%7D)
which represent the equations of lines SR and QS in that exact order.
Like before, we'll apply substitution to solve for x.
x-2y+12 = 0
x-2(2x)+12 = 0
x-4x+12 = 0
-3x+12 = 0
12 = 3x
12/3 = x
4 = x
x = 4
Which leads to y = 2x = 2*4 = 8
Point S is located at (4,8)
Point R is given to be located at (6,9)
Apply the distance formula for these two points to find the length of SR (aka the distance from S to R).
![S = (x_1,y_1) = (4,8)\\\\R = (x_2,y_2) = (6,9)\\\\d = \text{Length of SR, aka distance from S to R}\\\\d = \sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2}\\\\d = \sqrt{(4-6)^2 + (8-9)^2}\\\\d = \sqrt{(-2)^2 + (-1)^2}\\\\d = \sqrt{4 + 1}\\\\d = \sqrt{5}\\\\d \approx 2.236068\\\\](https://tex.z-dn.net/?f=S%20%3D%20%28x_1%2Cy_1%29%20%3D%20%284%2C8%29%5C%5C%5C%5CR%20%3D%20%28x_2%2Cy_2%29%20%3D%20%286%2C9%29%5C%5C%5C%5Cd%20%3D%20%5Ctext%7BLength%20of%20SR%2C%20aka%20distance%20from%20S%20to%20R%7D%5C%5C%5C%5Cd%20%3D%20%5Csqrt%7B%28x_1%20-%20x_2%29%5E2%20%2B%20%28y_1%20-%20y_2%29%5E2%7D%5C%5C%5C%5Cd%20%3D%20%5Csqrt%7B%284-6%29%5E2%20%2B%20%288-9%29%5E2%7D%5C%5C%5C%5Cd%20%3D%20%5Csqrt%7B%28-2%29%5E2%20%2B%20%28-1%29%5E2%7D%5C%5C%5C%5Cd%20%3D%20%5Csqrt%7B4%20%2B%201%7D%5C%5C%5C%5Cd%20%3D%20%5Csqrt%7B5%7D%5C%5C%5C%5Cd%20%5Capprox%202.236068%5C%5C%5C%5C)
The exact length is
and that approximates to roughly 2.236068 units.
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<h3>
Answer:</h3>
Exact length =
units
Approximate length = 2.236068 units