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Find the equation to the circle \(x^{2} + y^{2} - 4x - 2y = ...

Find the equation to the circle x2+y24x2y=0 at the point (1, 3).

A.

2y - x -5 = 0

B.

2y + x - 5 = 0

C.

2y + x + 5 = 0

D.

2y - x + 5 = 0

Correct answer is A

We are given the equation x2+y24x2y=0

y=x2+y24x2y

Using the method of implicit differentiation, 

dydx=2x+2ydydx42dydx

For the tangent, dydx=0,

(2y - 2)\frac{\mathrm d y}{\mathrm d x} = 4 - 2x \implies \frac{\mathrm d y}{\mathrm d x} = \frac{4 - 2x}{2y - 2}

At (1, 3), \frac{\mathrm d y}{\mathrm d x} = \frac{4 - 2(1)}{2(3) - 2} = \frac{2}{4} = \frac{1}{2}

Equation: \frac{y - 3}{x - 1} = \frac{1}{2} \implies 2y - 6 = x - 1

= 2y - x - 6 + 1 = 2y - x - 5 = 0