Rei A.

asked • 11/20/21

analytic geometry - alegbra problem, tysm :)

Solve the system ALGEBRAICALLY: 


<math xmlns="http://www.w3.org/1998/Math/MathML" display="block" data-is-equatio="1" data-latex="\begin{cases}\frac{\left(x-4\right)^2}{25}+\frac{\left(y+2\right)^2}{9}=1\\&#13;&#10;-\frac{\left(y+2\right)^2}{16}+\frac{\left(x-4\right)^2}{25}=1\end{cases}"><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">{</mo><mtable columnalign="left left" columnspacing="1em" rowspacing=".2em"><mtr><mtd><mfrac><msup><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mi>x</mi><mo>−</mo><mn>4</mn><mo data-mjx-texclass="CLOSE">)</mo></mrow><mn>2</mn></msup><mn>25</mn></mfrac><mo>+</mo><mfrac><msup><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mi>y</mi><mo>+</mo><mn>2</mn><mo data-mjx-texclass="CLOSE">)</mo></mrow><mn>2</mn></msup><mn>9</mn></mfrac><mo>=</mo><mn>1</mn></mtd></mtr><mtr><mtd><mo>−</mo><mfrac><msup><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mi>y</mi><mo>+</mo><mn>2</mn><mo data-mjx-texclass="CLOSE">)</mo></mrow><mn>2</mn></msup><mn>16</mn></mfrac><mo>+</mo><mfrac><msup><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mi>x</mi><mo>−</mo><mn>4</mn><mo data-mjx-texclass="CLOSE">)</mo></mrow><mn>2</mn></msup><mn>25</mn></mfrac><mo>=</mo><mn>1</mn></mtd></mtr></mtable><mo data-mjx-texclass="CLOSE" fence="true" stretchy="true" symmetric="true"/></mrow></math>


Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="block" data-is-equatio="1" data-latex="\left(a,b\right)"><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo data-mjx-texclass="CLOSE">)</mo></mrow></math> be an element of the solution set of the system above. Find the equation of the line containing <math xmlns="http://www.w3.org/1998/Math/MathML" display="block" data-is-equatio="1" data-latex="\left(-5,4\right)"><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mo>−</mo><mn>5</mn><mo>,</mo><mn>4</mn><mo data-mjx-texclass="CLOSE">)</mo></mrow></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="block" data-is-equatio="1" data-latex="\left(a,b\right)"><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo data-mjx-texclass="CLOSE">)</mo></mrow></math> for all such <math xmlns="http://www.w3.org/1998/Math/MathML" display="block" data-is-equatio="1" data-latex="\left(a,b\right)"><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo data-mjx-texclass="CLOSE">)</mo></mrow></math>.


I have (a,b) = (x,y) = (9,-2) or (-1, -2). Should I use the Slope Intercept Form to get the equation of the line? Does this mean that there are 2 equations of the line in this problem? Thank you!

Doug C.

That would be my interpretation, i.e. find equation... for all such (a,b). Since there are two points (a,b) there will be two equations. Find the slope between (-5,4) and each (a,b). Then use point-slope or slope-intercept to write the equations. I prefer to use point-slope and then convert to slope-intercept. y-4 = slopeFound(x+5)
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11/20/21

1 Expert Answer

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Yefim S. answered • 11/20/21

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We sabtract these equations: (y+ 2)2/9 + (y+ 2)2/16 = 0; (y + 2)2 = 0; y = - 2; (x - 4)2 = 25; x - 4 = ±5;

x = 9 or x = -1. We have 2 elements: (9, - 2) or (-1, - 2).

Line through (- 5, 4) and (9, - 2): slope m = (- 2 - 4)/(9 + 5) = - 3/7; y = 4 - 3/7(x + 5); y = - 3/7x + 13/7

Line through (- 5, 4) and (-1, - 2): slope m = (- 2 - 4)/(- 1 + 5) = - 3/2; y = 4 - 3/2(x + 5), y = - 3/2x - 7/2

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