Any solution anyone?

Need Help !!

#Geometry

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`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
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Comments

Im getting the answer as

$\frac{\pi ab}{4\left(a^2+b^2\right)}:0.5$

So, first, the area of the rhombus is $\frac{1}{2}ab$

Now, to find the area of the circle, we can first find the radius of the circle.

Let the center of the circle be at coordinate $(0,0)$ . and one of the rhombus's side be described with the equation $y=\frac{a}{b}\left(x-\frac{b}{2}\right)+a$

The radius will be the shortest distance between the center of the circle and the side of the rhombus. Finding that distance, we get $r^{2}=\frac{a^2b^2}{4\left(a^2+b^2\right)}$ The area of the circle is therefore $\ \pi \times \frac{a^2b^2}{4\left(a^2+b^2\right)}$

Hence, the ratio is $\boxed{\frac{\pi ab}{4\left(a^2+b^2\right)}:0.5}$

Julian Poon - 5 years, 9 months ago

You can write it as:

$\frac{\pi ab}{2(a^{2}+b^{2})}$

Syed Baqir - 5 years, 9 months ago

Can you elaborate more how you succeeded in finding the radius of circle,

It will be helpful if you add more lines , Thanks , (Upvoted)

Syed Baqir - 5 years, 9 months ago

Let the x-coordinate of the point where the circle will touch the side be $x_{0}$ . Now, for a certain point on the side of the rhombus, the coordinates will be $\left(k,\frac{a}{b}\left(k-\frac{b}{2}\right)+a\right)$ . The distance between that point and the origin can thus be found using Pythagoras theorem and is given as:

$\sqrt{k^2+\left(\frac{a}{b}\left(k-\frac{b}{2}\right)+a\right)^2}$

When $k=x_{0}$ , the above equation would be minimised. Note that $\sqrt{x_{0}^2+\left(\frac{a}{b}\left(x_{0}-\frac{b}{2}\right)+a\right)^2}=r$ , where r is the radius of the circle.

In order to find the minimum, we can find the minimum of $k^2+\left(\frac{a}{b}\left(k-\frac{b}{2}\right)+a\right)^2$ , where $x_{0}^2+\left(\frac{a}{b}\left(x_{0}-\frac{b}{2}\right)+a\right)^2=r^{2}$

All we have to do to find $r^{2}$ is to simplify the quadratic and find the minimum.

This might help.

Julian Poon - 5 years, 9 months ago

@Julian Poon – Thank you very much :D

By the way the Co-ordinate is too complicated to spot !!

Syed Baqir - 5 years, 9 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}`	$\boxed{123}$