# Thread:

1. ## Geometrical Properties of Ellipses

Hi.

1. If M and M' are the points where the tangent at P on the ellipse meets the tangents at the ends of the major axis, then MM' subtends a right angle at either focus. Show that the line y = mx + c touches the ellipse x^2/a^2 + y^2/b^2 = 1 when c = +/- sqrt[a^2m^2+b^2]

2. The normal to the ellipse x^2/a^2 + y^2/b^2 = 1 at P(x1, y1) meets the x-axis in N and the Y-axis in G. Prove PN/NG = (1-e^2)/e^2.

3. Show that the semi-latus rectum of the ellipse (acosx, bsiny) is of length b^2/a.

4. The extremities of any diameter of an ellipse are L, L' and P is any point on the ellipse. Show that the product of the gradients of the chords LP and L'P are constant.

2. Hi

The line $y = mx+c$ touches the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ when

$\frac{x^2}{a^2} + \frac{\left(mx+c\right)^2}{b^2} = 1$ has 1 solution

$\left(\frac{1}{a^2}+\frac{m^2}{b^2}\right)x^2 + \frac{2mc}{b^2} x + \frac{c^2}{b^2} - 1 = 0$ has 1 solution

therefore when the discriminant is equal to 0 which gives $c^2 = a^2m^2 + b^2$ after simplification
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 5, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.8356741070747375, "perplexity_flag": "middle"}