<PRE><B>Question 207882</B>
{{{graph(600,600,-10,10,-10,10,(-3/4)*x+(9/2),(4/3)*x-8,sqrt(25-(x-3)^2)-4,-sqrt(25-(x-3)^2)-4)}}}
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your answer is k = 4.5
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derivation of your answer is shown below:
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the equation of the line tangent to the circle is y = -(3/4)*x + k
we&#39;ll call that line P
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this line will have to be perpendicular to the radius of the circle at that point.
we&#39;ll call that line R.
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the slope of the line R must be equal to the negative reciprocal of the slope of the line P because the radius of the circle and the line tangent to it are perpendicular to each other at that point.
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the slope of line P is -(3/4).
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the slope of line R must be (4/3) which is the negative reciprocal of (-3/4)
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we know that the radius must go through the center of the circle.  This means that line R must go through the center of the circle.
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we know that the center of the circle is (3,-4) because the formula of the circle tells us so.
general formula of a circle is (x-h)^2 + (y-k)^2 = r^2 where (h,k) are the coordinates of the center of the circle.
since -h = -3, then h = 3
since -k = 4, then k = -4
center of the circle is (3,-4)
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general form of the equation of line R is y = (4/3)*x + b
where (4/3) is the slope and b is the y intercept.
to find the y intercept of that line, we plug in a value for one of the coordinates of that line and solve for b.
the center of the circle is one such point.
general form of line R becomes:
-4 = (4/3)*3 + b after we replace y with -4 and x with 3
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solving for b, we get b = -8
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equation of line R is y = (4/3)*x - 8
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we need to find the intersection of line R with the circle because that is also the point where line R will intersect with line P.
once we find that point, we can then solve the equation for line P which will result in allowing us to find k.
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we know the equation for line R.
we know the equation for the circle.
we can substitute the y value in the equation of the circle by the y value in the equation of line R.  this will allow us to find the x value of the intersection of the line R with the equation of the circle.
once we find x, we can solve for y.
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here&#39;s how it was done.
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the equation of the circle is (x-3)^2 + (y+4)^2 = 25
the equation of line R is y = (4/3)*x - 8
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since (x-3)^2 equals x^2 - 6x + 9
and since (y+4)^2 equals y^2 + 8y + 16
then the equation of the circle expands to:
x^2 - 6x + 9 + y^2 +8y + 16 = 25
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since we know that y = (4/3)*x - 8, we can substitute for y in this equation.
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8*y becomes 8*(4/3)*x - 64
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y^2 becomes ((4/3)*x -8) * ((4/3)*x - 8) which becomes:
(16/9)*x^2 - 8*(4/3)*x - 8*(4/3)*x + 64 which becomes:
(16/9)*x^2 - 16*(4/3)*x + 64
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y^2 + 8y becomes:
(16/9)*x^2 - 16*(4/3)*x + 64 + 8*(4/3)*x - 64 which becomes:
(16/9)*x^2 - 8*(4/3)*x
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the equation of the circle becomes:
x^2 - 6x + 9 + (16/9)*x^2 - 8*(4/3)*x + 16 = 25
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in order to remove the denominators, we multiply both sides of this equation by 9 to get:
9x^2 - 54x + 81 + 16x^2 - 96x + 144 = 225
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combining like terms we get:
25x^2 - 150x + 225 = 225
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subtracting 225 from both sides of this equation gets:
25x^2 - 150x = 0
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dividing both sides of this equation by 25 gets:
x^2 - 6x = 0
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factoring out the x gets:
x * (x-6) = 0
which results in:
x = 0
or:
x = 6
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as shown on the graph, both these values for x are good.
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solving for y using line R and the value of x = 6 gets:
y = (4/3)*6 - 8
which results in:
y = 0
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solving for y using the original equation of the circle and the value of x = 6 gets:
(6-3)^2 + (y+4)^2 = 25 which becomes:
9 + y^2 + 8y + 16 = 25 which becomes:
y^2 + 8y = 0 which becomes:
y * (y-8) = 0 which becomes:
y = 0
or
y = -8
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when x = 6, y = 0 is good.
when x = 0, y = -8 is good.
this can be seen from the graph.
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we now have the intersection of line R with the equation of the circle.
we also have the intersection of line R with line P since both line R and line P intersect with the equation of the circle at that point.
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now we have a point for line P that we can use to solve that equation.
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the equation for line P is y = (-3/4)*x + k
if we plug in the value of 6 for x and the value of 0 for y, the equation becomes:
0 = (-3/4)*(6) + k
this results in:
k = (3/4)*(6) which results in:
k = 4.5
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