• :00Days
• :00Hours
• :00Mins
• 00Seconds
A new era for learning is coming soon

Suggested languages for you:

Americas

Europe

Q. 61

Expert-verified
Found in: Page 773

### Calculus

Book edition 1st
Author(s) Peter Kohn, Laura Taalman
Pages 1155 pages
ISBN 9781429241861

# Consider the hyperbola with equation $\frac{{x}^{2}}{{A}^{2}}-\frac{{y}^{2}}{{B}^{2}}=1.$ Let F be the focus with coordinates $\left(\sqrt{{A}^{2}+{B}^{2}},0\right).$ Let $e=\frac{\sqrt{{A}^{2}+{B}^{2}}}{A}$ and l be the vertical line with equation $x=\frac{A}{e}.$ Show that for any point P on the hyperbola, $\frac{FP}{DP}=e,$ where D is the point on l closest to P.

The given statement is proved.

See the step by step solution

## Step 1. Given Information.

The given hyperbola equation is $\frac{{x}^{2}}{{A}^{2}}-\frac{{y}^{2}}{{B}^{2}}=1.$ and the coordinates of focus are $\left(\sqrt{{A}^{2}+{B}^{2}},0\right).$

## Step 2. Showing.

Let P be the point $\left(x,y\right)$ on the hyperbola.

So, the distance between the points $\left(x,y\right)\left(\sqrt{{A}^{2}+{B}^{2}},0\right).$

By using the formula for the distance $\sqrt{{\left({x}_{1}-{x}_{2}\right)}^{2}+{\left({y}_{1}-{y}_{2}\right)}^{2}}.$

$\mathrm{Here},{x}_{1}=x,{y}_{1}=y,{x}_{2}=\sqrt{{A}^{2}-{B}^{2}},{y}_{2}=0\phantom{\rule{0ex}{0ex}}FP=\sqrt{{\left(x-\left(\sqrt{{A}^{2}-{B}^{2}}\right)\right)}^{2}+\left(y-0{\right)}^{2}}\phantom{\rule{0ex}{0ex}}FP=\sqrt{\left({x}^{2}+{A}^{2}-{B}^{2}-2x\sqrt{{A}^{2}-{B}^{2}}\right)+{y}^{2}}\phantom{\rule{0ex}{0ex}}FP=\sqrt{\left({x}^{2}+{A}^{2}-{B}^{2}-2x\sqrt{{A}^{2}-{B}^{2}}\right)-{B}^{2}+\frac{{B}^{2}{x}^{2}}{{A}^{2}}}\phantom{\rule{0ex}{0ex}}\left[\mathrm{sin}ce,\frac{{x}^{2}}{{A}^{2}}-\frac{{y}^{2}}{{B}^{2}}=1⇒{y}^{2}=\frac{{B}^{2}{x}^{2}}{{A}^{2}}-{B}^{2}\right]\phantom{\rule{0ex}{0ex}}FP=\sqrt{{x}^{2}\left(1+\frac{{B}^{2}}{{A}^{2}}\right)+{A}^{2}-2x\sqrt{{A}^{2}+{B}^{2}}}\phantom{\rule{0ex}{0ex}}FP=\sqrt{\left(\frac{{A}^{2}+{B}^{2}}{{A}^{2}}\right)\left({x}^{2}-2x\sqrt{{A}^{2}+{B}^{2}}\frac{{A}^{2}}{{A}^{2}+{B}^{2}}+{A}^{2}·\frac{{A}^{2}}{{A}^{2}+{B}^{2}}\right)}\phantom{\rule{0ex}{0ex}}FP=\sqrt{\frac{{A}^{2}+{B}^{2}}{{A}^{2}}}\sqrt{\left({x}^{2}-2x\sqrt{{A}^{2}+{B}^{2}}\frac{{A}^{2}}{{\left(\sqrt{{A}^{2}+{B}^{2}}\right)}^{2}}+{A}^{2}·\frac{{A}^{2}}{{\left(\sqrt{{A}^{2}+{B}^{2}}\right)}^{2}}\right)}\phantom{\rule{0ex}{0ex}}FP=\sqrt{\frac{{A}^{2}+{B}^{2}}{{A}^{2}}}\sqrt{\left({x}^{2}-2x\frac{{A}^{2}}{\left(\sqrt{{A}^{2}+{B}^{2}}\right)}+{\left(\frac{{A}^{2}}{\left(\sqrt{{A}^{2}+{B}^{2}}\right)}\right)}^{2}\right)}\phantom{\rule{0ex}{0ex}}FP=\sqrt{\frac{{A}^{2}+{B}^{2}}{{A}^{2}}}\left(x-\frac{{A}^{2}}{\sqrt{{A}^{2}+{B}^{2}}}\right)\phantom{\rule{0ex}{0ex}}FP=e\left(x-\frac{A}{e}\right)\left[since\frac{\sqrt{{A}^{2}+{B}^{2}}}{A}=e\right]\phantom{\rule{0ex}{0ex}}\frac{FP}{DP}=\frac{e\left(x-\frac{A}{e}\right)}{x-\frac{A}{e}}\left[\mathrm{sin}ceFP=e\left(x-\frac{A}{e}\right),DP=x-\frac{A}{e}\right]\phantom{\rule{0ex}{0ex}}\frac{FP}{DP}=e$uncaught exception: Invalid chunkin file: /var/www/html/integration/lib/com/wiris/plugin/impl/HttpImpl.class.php line 68#0 /var/www/html/integration/lib/php/Boot.class.php(769): com_wiris_plugin_impl_HttpImpl_1(Object(com_wiris_plugin_impl_HttpImpl), NULL, 'http://www.wiri...', 'Invalid chunk') #1 /var/www/html/integration/lib/haxe/Http.class.php(532): _hx_lambda->execute('Invalid chunk') #2 /var/www/html/integration/lib/php/Boot.class.php(769): haxe_Http_5(true, Object(com_wiris_plugin_impl_HttpImpl), Object(com_wiris_plugin_impl_HttpImpl), Array, Object(haxe_io_BytesOutput), true, 'Invalid chunk') #3 /var/www/html/integration/lib/com/wiris/plugin/impl/HttpImpl.class.php(30): _hx_lambda->execute('Invalid chunk') #4 /var/www/html/integration/lib/haxe/Http.class.php(444): com_wiris_plugin_impl_HttpImpl->onError('Invalid chunk') #5 /var/www/html/integration/lib/haxe/Http.class.php(458): haxe_Http->customRequest(true, Object(haxe_io_BytesOutput), Object(sys_net_Socket), NULL) #6 /var/www/html/integration/lib/com/wiris/plugin/impl/HttpImpl.class.php(43): haxe_Http->request(true) #7 /var/www/html/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(268): com_wiris_plugin_impl_HttpImpl->request(true) #8 /var/www/html/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(307): com_wiris_plugin_impl_RenderImpl->showImage('d5a92cf1d81b51b...', NULL, Object(PhpParamsProvider)) #9 /var/www/html/integration/createimage.php(17): com_wiris_plugin_impl_RenderImpl->createImage('

## Step 3. Showing.

By proceeding with the calculation further,

$FP=\sqrt{\left(\frac{{A}^{2}+{B}^{2}}{{A}^{2}}\right)\left({x}^{2}-2x\sqrt{{A}^{2}-{B}^{2}}\frac{{A}^{2}}{{A}^{2}+{B}^{2}}+{A}^{2}·\frac{{A}^{2}}{{A}^{2}+{B}^{2}}\right)}\phantom{\rule{0ex}{0ex}}FP=\sqrt{\frac{{A}^{2}+{B}^{2}}{{A}^{2}}}\sqrt{\left({x}^{2}-2x\sqrt{{A}^{2}+{B}^{2}}\frac{{A}^{2}}{{\left(\sqrt{{A}^{2}+{B}^{2}}\right)}^{2}}+{A}^{2}·\frac{{A}^{2}}{{\left(\sqrt{{A}^{2}+{B}^{2}}\right)}^{2}}\right)}\phantom{\rule{0ex}{0ex}}FP=\sqrt{\frac{{A}^{2}+{B}^{2}}{{A}^{2}}}\sqrt{\left({x}^{2}-2x\frac{{A}^{2}}{\left(\sqrt{{A}^{2}+{B}^{2}}\right)}+{\left(\frac{{A}^{2}}{\left(\sqrt{{A}^{2}+{B}^{2}}\right)}\right)}^{2}\right)}\phantom{\rule{0ex}{0ex}}FP=\sqrt{\frac{{A}^{2}+{B}^{2}}{{A}^{2}}}\left(x-\frac{{A}^{2}}{\sqrt{{A}^{2}+{B}^{2}}}\right)\phantom{\rule{0ex}{0ex}}FP=e\left(x-\frac{A}{e}\right)\left[since\frac{\sqrt{{A}^{2}+{B}^{2}}}{A}=e\right]\phantom{\rule{0ex}{0ex}}\mathrm{Now},\phantom{\rule{0ex}{0ex}}\frac{FP}{DP}=\frac{e\left(x-\frac{A}{e}\right)}{x-\frac{A}{e}}\left[\mathrm{sin}ceFP=e\left(x-\frac{A}{e}\right),DP=x-\frac{A}{e}\right]\phantom{\rule{0ex}{0ex}}\frac{FP}{DP}=e$

Hence proved.