\n \nLet it meet the axes at and respectively. \nSolving equation with x-axis, i.e., we get \nSo, the co-ordinates of are \nSolving equation with y-axis, i.e., we get the co-ordinates of as \nIf the co-ordinates of the middle point of be , then \n and \nHence, \nAnd \nNow squaring and adding, we get \n \n
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The equation to normal at any point on the ellipse is \n \nLet it meet the axes at and respectively. \nSolving equation with x-axis, i.e., we get \nSo, the co-ordinates of are \nSolving equation with y-axis, i.e., we get the co-ordinates of as \nIf the co-ordinates of the middle point of be then \n and \nHence, and \nSquaring and adding, we get \n \n
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\n"},"comment":{"@type":"Comment","text":"Use the equation to normal at any point on the ellipse is and then find required mid point then eliminate variables."},"encodingFormat":"text/markdown","learningResourceType":"Practice problem","suggestedAnswer":[],"text":"The normal at meets the axes in and show that the loci of the middle point of and are respectively the ellipses \n and "},"name":"Quiz on The Ellipse","typicalAgeRange":"10-17","url":"https://www.embibe.com/questions/The-normal-at-P-meets-the-axes-in-G-and-g%3B-show-that-the-loci-of-the-middle-point-of-PG-and-Gg-are-respectively-the-ellipses%0A4x2a21%2Be22%2B4y2b2%3D1-and-a2x2%2Bb2y2%3D14a2-b22./EM2069530"}
S L Loney Solutions for Chapter: The Ellipse, Exercise 2: EXAMPLES XXXIII
Author:S L Loney
S L Loney Mathematics Solutions for Exercise - S L Loney Solutions for Chapter: The Ellipse, Exercise 2: EXAMPLES XXXIII
Attempt the practice questions on Chapter 10: The Ellipse, Exercise 2: EXAMPLES XXXIII with hints and solutions to strengthen your understanding. The Elements of COORDINATE GEOMETRY Part 1 Cartesian Coordinates solutions are prepared by Experienced Embibe Experts.
Questions from S L Loney Solutions for Chapter: The Ellipse, Exercise 2: EXAMPLES XXXIII with Hints & Solutions
If a number of ellipses be described having the same major axis, but a variable minor axis, prove that the tangents at the ends of their latus recta pass through one or other of two fixed points.
Prove that the sum of the eccentric angles of the extremities of a chord, which is drawn in a given direction, is constant, and equal to twice the eccentric angle of the point at which the tangent is parallel to the given direction.