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.

Did you find an error or inaccuracy? Feel free to write us. Thank you!

Tips for related online calculators

Are you looking for help with calculating roots of a quadratic equation?
Do you have a linear equation or system of equations and are looking for its solution? Or do you have a quadratic equation?
Do you want to convert time units like minutes to seconds?