Mister Exam

# Express x in terms of y where (x-17)^2+(y+26)^2=16(x−17)2+(y+26)2=16﻿.

The teacher will be very surprised to see your correct solution 😉

v

#### Numerical solution:

Do search numerical solution at [, ]

### The solution

You have entered [src]
        2           2
(x - 17)  + (y + 26)  = 16*(x - 17)*2 + (y + 26)*2
$$\left(x - 17\right)^{2} + \left(y + 26\right)^{2} = 2 \cdot 16 \left(x - 17\right) + 2 \left(y + 26\right)$$
Detail solution
Move right part of the equation to
left part with negative sign.

The equation is transformed from
$$\left(x - 17\right)^{2} + \left(y + 26\right)^{2} = 2 \cdot 16 \left(x - 17\right) + 2 \left(y + 26\right)$$
to
$$\left(- 2 \cdot 16 \left(x - 17\right) - 2 \left(y + 26\right)\right) + \left(\left(x - 17\right)^{2} + \left(y + 26\right)^{2}\right) = 0$$
Expand the expression in the equation
$$\left(- 2 \cdot 16 \left(x - 17\right) - 2 \left(y + 26\right)\right) + \left(\left(x - 17\right)^{2} + \left(y + 26\right)^{2}\right) = 0$$
$$x^{2} - 66 x + y^{2} + 50 y + 1457 = 0$$
This equation is of the form
a*x^2 + b*x + c = 0

A quadratic equation can be solved
using the discriminant.
The roots of the quadratic equation:
$$x_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$x_{2} = \frac{- \sqrt{D} - b}{2 a}$$
where D = b^2 - 4*a*c - it is the discriminant.
Because
$$a = 1$$
$$b = -66$$
$$c = y^{2} + 50 y + 1457$$
, then
D = b^2 - 4 * a * c =

(-66)^2 - 4 * (1) * (1457 + y^2 + 50*y) = -1472 - 200*y - 4*y^2

The equation has two roots.
x1 = (-b + sqrt(D)) / (2*a)

x2 = (-b - sqrt(D)) / (2*a)

or
$$x_{1} = \frac{\sqrt{- 4 y^{2} - 200 y - 1472}}{2} + 33$$
$$x_{2} = 33 - \frac{\sqrt{- 4 y^{2} - 200 y - 1472}}{2}$$
Rapid solution [src]
              _____________________________________________________________________                                                                                  _____________________________________________________________________
/                                                                   2     /     /                                    2        2              \\        /                                                                   2     /     /                                    2        2              \\
4 /                             2   /         2        2              \      |atan2\-50*im(y) - 2*im(y)*re(y), -368 + im (y) - re (y) - 50*re(y)/|     4 /                             2   /         2        2              \      |atan2\-50*im(y) - 2*im(y)*re(y), -368 + im (y) - re (y) - 50*re(y)/|
x1 = 33 - \/   (-50*im(y) - 2*im(y)*re(y))  + \-368 + im (y) - re (y) - 50*re(y)/  *cos|-------------------------------------------------------------------| - I*\/   (-50*im(y) - 2*im(y)*re(y))  + \-368 + im (y) - re (y) - 50*re(y)/  *sin|-------------------------------------------------------------------|
\                                 2                                 /                                                                                  \                                 2                                 /
$$x_{1} = - i \sqrt[4]{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 50 \operatorname{im}{\left(y\right)}\right)^{2} + \left(- \left(\operatorname{re}{\left(y\right)}\right)^{2} - 50 \operatorname{re}{\left(y\right)} + \left(\operatorname{im}{\left(y\right)}\right)^{2} - 368\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 50 \operatorname{im}{\left(y\right)},- \left(\operatorname{re}{\left(y\right)}\right)^{2} - 50 \operatorname{re}{\left(y\right)} + \left(\operatorname{im}{\left(y\right)}\right)^{2} - 368 \right)}}{2} \right)} - \sqrt[4]{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 50 \operatorname{im}{\left(y\right)}\right)^{2} + \left(- \left(\operatorname{re}{\left(y\right)}\right)^{2} - 50 \operatorname{re}{\left(y\right)} + \left(\operatorname{im}{\left(y\right)}\right)^{2} - 368\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 50 \operatorname{im}{\left(y\right)},- \left(\operatorname{re}{\left(y\right)}\right)^{2} - 50 \operatorname{re}{\left(y\right)} + \left(\operatorname{im}{\left(y\right)}\right)^{2} - 368 \right)}}{2} \right)} + 33$$
              _____________________________________________________________________                                                                                  _____________________________________________________________________
/                                                                   2     /     /                                    2        2              \\        /                                                                   2     /     /                                    2        2              \\
4 /                             2   /         2        2              \      |atan2\-50*im(y) - 2*im(y)*re(y), -368 + im (y) - re (y) - 50*re(y)/|     4 /                             2   /         2        2              \      |atan2\-50*im(y) - 2*im(y)*re(y), -368 + im (y) - re (y) - 50*re(y)/|
x2 = 33 + \/   (-50*im(y) - 2*im(y)*re(y))  + \-368 + im (y) - re (y) - 50*re(y)/  *cos|-------------------------------------------------------------------| + I*\/   (-50*im(y) - 2*im(y)*re(y))  + \-368 + im (y) - re (y) - 50*re(y)/  *sin|-------------------------------------------------------------------|
\                                 2                                 /                                                                                  \                                 2                                 /
$$x_{2} = i \sqrt[4]{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 50 \operatorname{im}{\left(y\right)}\right)^{2} + \left(- \left(\operatorname{re}{\left(y\right)}\right)^{2} - 50 \operatorname{re}{\left(y\right)} + \left(\operatorname{im}{\left(y\right)}\right)^{2} - 368\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 50 \operatorname{im}{\left(y\right)},- \left(\operatorname{re}{\left(y\right)}\right)^{2} - 50 \operatorname{re}{\left(y\right)} + \left(\operatorname{im}{\left(y\right)}\right)^{2} - 368 \right)}}{2} \right)} + \sqrt[4]{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 50 \operatorname{im}{\left(y\right)}\right)^{2} + \left(- \left(\operatorname{re}{\left(y\right)}\right)^{2} - 50 \operatorname{re}{\left(y\right)} + \left(\operatorname{im}{\left(y\right)}\right)^{2} - 368\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 50 \operatorname{im}{\left(y\right)},- \left(\operatorname{re}{\left(y\right)}\right)^{2} - 50 \operatorname{re}{\left(y\right)} + \left(\operatorname{im}{\left(y\right)}\right)^{2} - 368 \right)}}{2} \right)} + 33$$
x2 = i*((-2*re(y)*im(y) - 50*im(y))^2 + (-re(y)^2 - 50*re(y) + im(y)^2 - 368)^2)^(1/4)*sin(atan2(-2*re(y)*im(y) - 50*im(y, -re(y)^2 - 50*re(y) + im(y)^2 - 368)/2) + ((-2*re(y)*im(y) - 50*im(y))^2 + (-re(y)^2 - 50*re(y) + im(y)^2 - 368)^2)^(1/4)*cos(atan2(-2*re(y)*im(y) - 50*im(y), -re(y)^2 - 50*re(y) + im(y)^2 - 368)/2) + 33)