How do you determine circle, parabola, ellipse, or hyperbola from equation #x^2 + y^2 - 16x + 18y - 11 = 0#?

↳Redirected from "Suppose that I don't have a formula for #g(x)# but I know that #g(1) = 3# and #g'(x) = sqrt(x^2+15)# for all x. How do I use a linear approximation to estimate #g(0.9)# and #g(1.1)#?"
1 Answer
sente
Nov 30, 2015

The equation is of a circle of radius #sqrt(156)# centered at #(8, -9)#

Explanation:

Step 1: Group #x#'s and #y#'s

#x^2 - 16x + y^2 + 18y = 11#

Step 2: Complete the square for both #x# and #y#

#x^2 - 16x + 64 + y^2 + 18y + 81 = 11 + 64 + 81#

#=>(x - 8)^2 + (y + 9)^2 = 156#

Step 3: Compare to the standard forms of conic sections

![https://www.pinterest.com/pin/429953095650353121/](useruploads.socratic.org)

Note that the above equation matches the formula for a circle with #h = 8#, #k = -9#, and #r = sqrt(156)#

Thus the equation is of a circle of radius #sqrt(156)# centered at #(8, -9)#

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