2Thanks
• 1 Post By topsquark
• 1 Post By kalyanram

# Thread:

1. ## Tangent lines to an ellipse from an external point

Given an the equation of an ellipse, say $\frac{x^2}{9}+\frac{y^2}{16}=1$,
find the equations of the two lines tangent to this ellipse passing through the external point
P(5,6). Thank you in advance.

2. ## Re: Tangent lines to an ellipse from an external point

Originally Posted by Kaloda
Given an the equation of an ellipse, say $\frac{x^2}{9}+\frac{y^2}{16}=1$,
find the equations of the two lines tangent to this ellipse passing through the external point
P(5,6). Thank you in advance.
1. Solve the equation for y^2
2. Implicit differentiation

That gives the equation for the tangent lines.
3. Plug in point P.
4. What can you come up with?

-Dan

3. ## Re: Tangent lines to an ellipse from an external point

Compute the slope of the tangent at the point of contact say $(\alpha,\beta)$which is $\frac{dy}{dx} = \frac{-16\alpha}{9\beta}$. Now we have the slope of the line also as $\frac{6-\beta}{5-\alpha} = \frac{-16\alpha}{9\beta} \implies 16\alpha^2 + 9 \beta^2 = 9*6\beta+9*10\alpha$ from the equation of ellipse we have $16\alpha^2 + 9 \beta^2 = 9*16=9*6\beta+9*10\alpha \implies 5\alpha+3\beta=8$ substitute in $\frac{\alpha^2}{9} + \frac{\beta^2}{16} = 1$ and solve for $\beta$ you will get two points on the ellipse, and hence you can get two lines.

~Kalyan.
Attached Thumbnails

4. ## Re: Tangent lines to an ellipse from an external point

I had already figured it out but THANKS.
{"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": 8, "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": 0.8810870051383972, "perplexity_flag": "middle"}