C2 co-oridnate geomtery Watch

f45
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#1
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#1
the line y=0 is a tangent to the circle (x-8)^2+(y-a)^2=16 find the value of a
how would I go about doing this first I thought find where the intersect, so sub y=0 into the equation of the circles but that gave me
x^2-16x+48+a^2???
method plz, don't care about the answer
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qgujxj39
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#2
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What is the gradient of the line y = 0?

What does the fact that this is a tangent tell you that \frac{dy}{dx} equals for the circle at the point where y = 0?
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Raminder1992
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you know that at (8,0) is where the tangent is below the centre
You also know the radius is 4
therefore the y co-ordinate is 4 above (8,4)
therefore a = 4
i think
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sa55afras
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#4
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First I'm substituting, like you said:



(x-8)^2+(y-a)^2=16

\par (x - 8)^2+(-a)^2=16

\par (x - 8)^2+a^2=16

\par (x - 8)^2=16-a^2

So, we know there's only one intersection point, because it's a tangent. Therefore, we know (x-8)^2=16-a^2 (obtained above) has exactly one solution, regardless of the value of x. This means that 16-a^2=0 because, if 16-a^2<0 there would be no solutions and if 16-a^2>0 there would be two solutions.

Does that make sense? Hope it helped
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qgujxj39
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#5
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sa55afras' way is very good, actually. Raminder's is also good, although it assumes the circle is above y=0 rather than below.
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Raminder1992
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(Original post by tommm)
sa55afras' way is very good, actually. Raminder's is also good, although it assumes the circle is above y=0 rather than below.
that is true but the square root of 16 is plus or minus 4. So how would you get around this problem?
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qgujxj39
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(Original post by Raminder1992)
that is true but the square root of 16 is plus or minus 4. So how would you get around this problem?
If you instead say that the tangent is at the top of the circle, then the same method gets you a = -4.
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sa55afras
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(Original post by tommm)
If you instead say that the tangent is at the top of the circle, then the same method gets you a = -4.
In fact there are two circles for which the condition is true, right?
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qgujxj39
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(Original post by sa55afras)
In fact there are two circles for which the condition is true, right?
Yup.
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Raminder1992
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(Original post by tommm)
If you instead say that the tangent is at the top of the circle, then the same method gets you a = -4.
thanks
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f45
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#11
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umm the answer is 5 :s
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