A few properties for solving eclipse problems

For an eclipse centered at the origin $\dfrac { {x }^{ 2} }{ { a }^{ 2} } +\dfrac { {y }^{ 2} }{ { b}^{ 2} } =1$ :

Any point $P$ on the eclipse can be expressed as $(a\cos\theta ,b\sin\theta )$ (or $(a\sin\theta ,b\cos\theta )$ );
Equation of the line tangent to the eclipse through point P is $\dfrac {x\cos\theta }{a } +\dfrac {y\sin\theta }{b } =1$ ;
The distance from the origin to the line $ax + by = c$ is $\left|\dfrac { c }{ \sqrt { { a }^{2 }+{ b }^{ 2 } } } \right|$ ;

image credit: jatin yadav.

#Geometry

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Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

Wow nice! You should put them in the wiki here!

Pi Han Goh - 5 years, 6 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$