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steve816
 one year ago
Please help me ASAP!
How do I find the equation of the circle?
The circle is tangent to the line y = 2x + 5 with center (5, 5).
steve816
 one year ago
Please help me ASAP! How do I find the equation of the circle? The circle is tangent to the line y = 2x + 5 with center (5, 5).

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campbell_st
 one year ago
Best ResponseYou've already chosen the best response.1well there is a formula for the perpendicular distance from a point to a line... in this case the distance would be the radius...

campbell_st
 one year ago
Best ResponseYou've already chosen the best response.1so rewrite the line as 2x  y + 5 = 0 then the formula is \[d = \frac{2x  y + 5}{\sqrt{2^2 + (1)^2}}\] just substitute the point (5, 5) to find the radius

campbell_st
 one year ago
Best ResponseYou've already chosen the best response.1then the equation will be \[(x  h)^2 + (y  k)^2 = r^2\] (h, k) is the centre, you have (5, 5) and r is the radius, in your case use d. hope it makes sense

steve816
 one year ago
Best ResponseYou've already chosen the best response.0But how did you get the denominator for d?

campbell_st
 one year ago
Best ResponseYou've already chosen the best response.1well is the equation of the tangent is in standard form Ax + By + C = 0 the formula is \[d = \frac{ \left Ax + By + C \right}{\sqrt{A^2 + B^2}}\] the denominator is from pythagoras' theorem. so all you do is just substitute you value of x = 5 and y = 5 and you get \[d = \frac{\left 20 \right}{\sqrt{5}}\] so that's the radius... square the numerator and denominator and you get \[d^2 = (\frac{20}{\sqrt{5}})^2~~~or~~~~d^2 = \frac{400}{5}\] then simplify that. I did it quickly so just check the calculations

IrishBoy123
 one year ago
Best ResponseYou've already chosen the best response.1you can also tell that the normal to the circle at the intersection is of the form \(y = \frac{1}{2} x + c \) as it's slope will be 1/m for a tangent with slope m thusly, use the centre (5,5) of the circle to solve for c, as the normal also runs through the centre of the circle so \(5 = \frac{1}{2} (5) + c \). then solve \(\frac{1}{2} x + c\ = 2x + 5\) to find the point where the circle actually intersects the tangent line. the circle's radius will be the distance from that point to the centre of circle at (5,5). dw:1443694428546:dw
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