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For what value of k does the equation x^2+10x+y^2+16y-k=0 represent a circle of radius 6?

 
 Jan 14, 2021
 #1
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The equation of a circle is (x+a)^2+(y+b)^2=r^2

(x+a)^2 is equal to x^2+2ax+a^2

(y+a)^2 is equal to y^2+2by+b^2

The radius is 6, so right side of the equation is 36

So X^2+2ax+a^2+y^2+2by+y^2=36

a is 5 since 2ax=10x

b is 8 bc 2by is 16y

Substituting we get (x+5)^2+(y+8)^2=36

Expanding we get x^2+10x+25+y^2+16y+64

Subtract 36 from both sides

x^2+10x+y^2+16y-k=0

25+64-36=53

-k=53

k=-53

 
 Jan 14, 2021
 #2
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x^2 + 10x       + y^2 + 16 y        = k          put in standard form by completing the square for x and y

(x+5)2               + (y+8)2             =  k  + 25 + 64

 

    and   k + 25 + 64   = 36    (the radius2)         so k = -53

 
 Jan 14, 2021

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