Book edition 5th

Author(s) Otto Bretscher

Pages 442 pages

ISBN 9780321796974

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Short Answer

Find the circle that runs through the points (5,5),(4,6), and (6,2). Write your equation in the form $a + b x + c y + x^{2} + y^{2} = 0$ . Find the centre and radius of this circle.

The required equation is $x^{2} + y^{2} - 2 x - 4 y = 20$ , which is the circle with centre (1,2) and radius 5 that runs through the points (5,5),(4,6) and (6,2).

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Consider the points and substitute them in the equation

The centre and the radius of the circle of form $(x - a)^{2} + (y - b)^{2} = r^{2}$ is $(a, b)$ and r respectively.

Substitute the given points in the equation $a + b x + c y + x^{2} + y^{2} = 0$

$5^{2} + 5^{2} + a + (5) b + (5) c = 0 a + 5 b + 5 c = - 50$

$4^{2} + 6^{2} + a + (4) b + (6) c = 0 a + 4 b + 6 c = - 52$

$6^{2} + 2^{2} + a + (6) b + (2) c = 0 a + 6 b + 2 c = - 40$

Step 2: Rearrange the terms of the above equations

Consider the simplified equations.

$- 50 = a + 5 b + 5 c . . . . . . (1) - 52 = a + 4 b + 6 c . . . . . . (2) - 40 = a + 6 b + 2 c . . . . . . (3)$

Step 3: Solve the above equations (1), (2) and (3)

Arrange the equations (1), (2) and (3) into matrix form:

$(\begin{matrix} 1 & 5 & 5 \\ 1 & 4 & 6 \\ (1 & 6 & 2 \end{matrix}) (\begin{matrix} a \\ b \\ c \end{matrix}) = (\begin{matrix} - 50 \\ - 52 \\ - 40 \end{matrix})$

Upon solving the values of $a, b$ and are obtained as $a = - 20, b = - 2, c = - 4$

Substitute these values in the given equation of the circle.

$x^{2} + y^{2} + (- 20) + (- 2) x + (- 4) y = 0 x^{2} + y^{2} - 20 - 2 x - 4 y = 0$

Step 4: Rearrange the terms of the above equation

Rearrange the resulting equation into standard form of circle as:

$x^{2} + y^{2} - 20 - 2 x - 4 y = 0 x^{2} + y^{2} - 2 x - 4 y = 20 (x^{2} + 1 - 2 x) + (y^{2} + 4 - 4 y) = 20 + 1 + 4 (x - 1)^{2} + (y - 2)^{2} = 25$

Compare the above equation with the standard equation of the circle.

${(x - a)}^{2} + {(y - b)}^{2} = r^{2} \Leftrightarrow {(x - 1)}^{2} + {(y - 2)}^{2} = 5^{2} (a, b) = (1, 2) r = 5$

The required equation is $x^{2} + y^{2} - 2 x - 4 y = 20$ , which is the circle with centre (1,2) and radius 5 that runs through the points (5,5),(4,6) and (6,2).

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Linear Algebra With Applications

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Short Answer

Find the circle that runs through the points (5,5),(4,6), and (6,2). Write your equation in the form $a + b x + c y + x^{2} + y^{2} = 0$ . Find the centre and radius of this circle.

Step by Step Solution

Step 1: Consider the points and substitute them in the equation

Step 2: Rearrange the terms of the above equations

Step 3: Solve the above equations (1), (2) and (3)

Step 4: Rearrange the terms of the above equation

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Linear Algebra With Applications

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Short Answer

Find the circle that runs through the points (5,5),(4,6), and (6,2). Write your equation in the form a+bx+cy+x2+y2=0. Find the centre and radius of this circle.

Step by Step Solution

Step 1: Consider the points and substitute them in the equation

Step 2: Rearrange the terms of the above equations

Step 3: Solve the above equations (1), (2) and (3)

Step 4: Rearrange the terms of the above equation

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Find the circle that runs through the points (5,5),(4,6), and (6,2). Write your equation in the form $a + b x + c y + x^{2} + y^{2} = 0$ . Find the centre and radius of this circle.