Question 940984
Find the equation of a circle centre on the line y=2x+1 touching the y axis and passing through A(4,5)
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If by touching you mean tangent:
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Its center is equidistant from (4,5) and the y-axis.
The center is (h,k)
The distance to (4,5) = {{{sqrt((h-4)^2 + (k-5)^2)}}}
The distance to the y-axis = h
y = 2x + 1 --> k = 2h + 1
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{{{h^2 = (h-4)^2 + (k-5)^2}}}
{{{h^2 = h^2-8h+16 + k^2-10k+25}}}
{{{-8h+16 + k^2-10k+25 = 0}}}
Sub for k
{{{-8h+16 + (2h+1)^2-10(2h+1)+25 = 0}}}
{{{-8h+16 + 4h^2+4h+1-20h-10+25 = 0}}}
{{{4h^2 - 4h + 17 - 20h + 15 = 0}}}
{{{4h^2 - 24h + 32 = 0}}}
*[invoke solve_quadratic_equation 4,-24,32]
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h = x = 4  --> center @ (4,9)
{{{(x-4)^2 + (y-9)^2 = 16}}}
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x = 2 --> center @ (2,5)
{{{(x-2)^2 + (y-5)^2 = 4}}}