Lesson Derive an equation and show that the locus of points is an ellipse
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<H2>Derive an equation and show that the locus of points is an ellipse</H2> <H3>Problem 1</H3>What is the equation of the locus of the point which moves so that its distance from the point (2,0) is 2/3 its distance from the line y = 5 ? <B>Solution</B> <pre> Let (x,y) be the point of the locus. Then the distance to point (2,0) is d = {{{sqrt((x-2)^2+y^2)}}}, while the distance to the line y=5 is |y-5|. Equate the equal distances {{{sqrt((x-2)^2+y^2)}}} = {{{(2/3)*abs(y-5)}}}. It is the basic equation which translates the input into mathematical form. Next, square both sides of this equation and reduce it to the standard conic section equation. {{{(x-2)^2 + y^2}}} = {{{(4/9)*(y-5)^2}}} {{{x^2 - 4x + 4 + y^2}}} = {{{(4/9)*(y^2 - 10y + 25)}}} {{{9x^2 - 36x + 36 + 9y^2}}} = {{{4y^2 - 40y + 100}}} {{{9x^2 - 36x}}} + {{{5y^2 + 40y}}} = 64 {{{9*(x^2 - 4x + 4)}}} + {{{5*(y^2 + 8y + 16)}}} = 64 + 36 + 80 {{{9*(x-2)^2}}} + {{{5*(y+4)^2}}} = 180. {{{(x-2)^2/20}}} + {{{(y+4)^2/36}}} = 1 It is the standard form equation of the ellipse centered at the point (2,-4) and the major semi-axis {{{sqrt(36)}}} = 6 (vertical) and the minor semi-axis {{{sqrt(20)}}} = {{{2*sqrt(5)}}} (horizontal). </pre> My other lessons in this site on deducing equations for locuses of points are - <A HREF=https://www.algebra.com/algebra/homework/Quadratic-relations-and-conic-sections/Derive-an-equation-that-a-locus-of-points-is-a-STRAIGHT-LINE.lesson>Derive an equation and show that the locus of points is a straight line</A> - <A HREF=https://www.algebra.com/algebra/homework/Quadratic-relations-and-conic-sections/Derive-an-equation-that-a-locus-of-points-is-a-CIRCLE.lesson>Derive an equation and show that the locus of points is a circle</A> - <A HREF=https://www.algebra.com/algebra/homework/Quadratic-relations-and-conic-sections/Derive-an-equation-and-show-that-the-locus-of-points-is-a-HYPERBOLA.lesson>Derive an equation and show that the locus of points is a hyperbola</A> - <A HREF=https://www.algebra.com/algebra/homework/Quadratic-relations-and-conic-sections/Derive-an-equation-and-show-that-the-locus-of-points-is-a-PARABOLA.lesson>Derive an equation and show that the locus of points is a parabola</A> - <A HREF=https://www.algebra.com/algebra/homework/Quadratic-relations-and-conic-sections/OVERWIEW-of-lessons-on-deducing-equations--for-locuses-of-points.lesson>OVERWIEW of lessons on deducing equations for locuses of points</A> Use this file/link <A HREF=https://www.algebra.com/algebra/homework/complex/ALGEBRA-II-YOUR-ONLINE-TEXTBOOK.lesson>ALGEBRA-II - YOUR ONLINE TEXTBOOK</A> to navigate over all topics and lessons of the online textbook ALGEBRA-II.
