Q36E Consider three distinct points [... [FREE SOLUTION]

91影视

Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 6: Q36E (page 290)

Consider three distinct points $[\begin{matrix} a_{1} \\ a_{2} \end{matrix}] [\begin{matrix} b_{1} \\ b_{2} \end{matrix}] [\begin{matrix} c_{1} \\ c_{2} \end{matrix}]$ in the plane. Describe the set of all points $[\begin{matrix} x_{1} \\ x_{2} \end{matrix}]$ satisfying the equation
$d e t [\begin{matrix} 1 & 1 & 1 & 1 \\ x_{1} & a_{1} & b_{1} & c_{1} \\ x_{2} & a_{2} & b_{2} & c_{2} \\ x_{1}^{2} + x_{2}^{2} & a_{1}^{2} + a_{2}^{2} & b_{1}^{2} + b_{2}^{2} & c_{1}^{2} + c_{2}^{2} \end{matrix}] = 0$

Short Answer

Expert verified

Therefore, the three given points are collinear, the solution is a line through those points; otherwise, it is a circle, also through all three points.

Step by step solution

Step by Step Solution: Step 1: Given

Consider three distinct points,

$[\begin{matrix} a_{1} \\ a_{2} \end{matrix}] [\begin{matrix} b_{1} \\ b_{2} \end{matrix}] [\begin{matrix} c_{1} \\ c_{2} \end{matrix}]$

To solve

By Knowing that a matrix is non-invertible if and only if its rows are linearly dependent, we have:

$x_{1}^{2} + x_{2}^{2} = 伪 + 尾 x_{1} + 纬 x_{2} a_{1}^{2} + a_{2}^{2} = 伪 + 尾 a_{1} + 纬 a_{2} b_{1}^{2} + b_{2}^{2} = 伪 + 尾 b_{1} + 纬 b_{2} c_{1}^{2} + c_{2}^{2} = 伪 + 尾 c_{1} + 纬 c_{2}$

This means that, if the three given points are collinear, the solution is a line through those points; otherwise, it is a circle, also through all three points.

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Consider three distinct points [a1a2][b1b2][c1c2] in the plane. Describe the set of all points[x1x2] satisfying the equationdet[1111x1a1b1c1x2a2b2c2x12+x22a12+a22b12+b22c12+c22]=0

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