Vertex of the rectangle

Determine the coordinates of the vertex of the rectangle inscribed in the circle x²+y² -2x-4y-20=0 if you know that one of its sides lies on the line p: x+2y=0

Correct answer:

A = (-4,2)
B = (4,-2)
C = (6,2)
D = (-2,6)

Step-by-step explanation:

x^{2} + y^{2} - 2 x - 4 y - 20 = 0 (x - 1)^{2} + (y - 2)^{2} = 25 = 5^{2} S = (1, 2) (- 2 y)^{2} + y^{2} - 2 \cdot (- 2 y) - 4 y - 20 = 0 (- 2 y)^{2} + y^{2} - 2 \cdot (- 2 y) - 4 y - 20 = 0 5 y^{2} - 20 = 0 5 . . . prime number 0 20 = 2^{2} \cdot 5 GCD (5, 0, 20) = 5 y^{2} - 4 = 0 y_{1, 2} = \pm 4 = \pm 2 y_{1} = 2 y_{2} = - 2 x_{1} = - 2 \cdot y_{1} = - 2 \cdot 2 = - 4 x_{2} = - 2 \cdot y_{2} = - 2 \cdot (- 2) = 4 A = (x_{1}, y_{1}) A = (- 4, 2)

Our quadratic equation calculator calculates it.

B = (x_{2}, y_{2}) B = (4, - 2)

q = A S A_{y} = a \cdot A_{x} + b S_{y} = a \cdot S_{x} + b 2 = a \cdot (- 4) + b 2 = a \cdot (- 1) + b 4 a - b = - 2 a - b = - 2 R o w 2 - \frac{1}{4} \cdot R o w 1 \to R o w 2 4 a - b = - 2 - 0.75 b = - 1.5 b = \frac{- 1 . 5}{- 0 . 7 5} = 2 a = \frac{- 2 + b}{4} = \frac{- 2 + 2}{4} = 0 a = 0 b = 2 (w)^{2} + 2^{2} - 2 \cdot (w) - 4 \cdot 2 - 20 = 0 (w)^{2} + 2^{2} - 2 \cdot (w) - 4 \cdot 2 - 20 = 0 w^{2} - 2 w - 24 = 0 a = 1; b = - 2; c = - 24 D = b^{2} - 4 a c = 2^{2} - 4 \cdot 1 \cdot (- 24) = 100 D > 0 w_{1, 2} = \frac{- b \pm D}{2 a} = \frac{2 \pm 1 0 0}{2} w_{1, 2} = \frac{2 \pm 1 0}{2} w_{1, 2} = 1 \pm 5 w_{1} = 6 w_{2} = - 4 C = (w_{1}, b) C = (6, 2)

Our quadratic equation calculator calculates it.

t = B S B_{y} = c \cdot B_{x} + d S_{y} = c \cdot S_{x} + d 4 = c \cdot (- 2) + d 2 = c \cdot (- 1) + d 2 c - d = - 4 c - d = - 2 R o w 2 - \frac{1}{2} \cdot R o w 1 \to R o w 2 2 c - d = - 4 - 0.5 d = 0 d = \frac{0}{- 0 . 5} = - 0 c = \frac{- 4 + d}{2} = \frac{- 4}{2} = - 2 c = - 2 d = - 0 t : x = - 2 (- 2)^{2} + v^{2} - 2 \cdot (- 2) - 4 \cdot v - 20 = 0 (- 2)^{2} + v^{2} - 2 \cdot (- 2) - 4 \cdot v - 20 = 0 v^{2} - 4 v - 12 = 0 a = 1; b = - 4; c = - 12 D = b^{2} - 4 a c = 4^{2} - 4 \cdot 1 \cdot (- 12) = 64 D > 0 v_{1, 2} = \frac{- b \pm D}{2 a} = \frac{4 \pm 6 4}{2} v_{1, 2} = \frac{4 \pm 8}{2} v_{1, 2} = 2 \pm 4 v_{1} = 6 v_{2} = - 2 D = (c, v_{1}) D = (- 2, 6)

Our quadratic equation calculator calculates it.

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