## Wednesday, December 5, 2012

### Determine whether an ellipse intersect a horizontal or vertical line

Assume an ellipse of width $$\sigma$$ and length $$\kappa \sigma$$ is centered at $$(x_0, y_0)$$, and has angle $$\theta_0$$ with the $$x$$-axis. How do we determine whether it intersects a horizontal line or a vertical line?

It turns out the criteria is very simple
• The ellipse intersects a horizontal line $$y = y_1$$ if and only if the following equation hods: $\triangle_1 = \sigma^2 \left(\cos^2(\theta_0) + \kappa^2 \sin^2(\theta_0)\right) - (y_1-y_0)^2 \geq 0$
• The ellipse intersects a vertical line $$x = x_1$$ if and only if the following equation hods: $\triangle_2 = \sigma^2 \left(\sin^2(\theta_0) + \kappa^2 \cos^2(\theta_0)\right) - (x_1-x_0)^2 \geq 0$

#### 1 comment:

albert said...

hi i have a hard question if you help me i will be very happy; In a country, there are four types of banknotes: 5, 10, 20 and 25 unit values. Each banknote has a serial number with more than four digits. A banknote is chosen randomly. What is the probability that the sum of the four last digits of the serial number is bigger than the banknote unit value?

Note:Four banknote types, and all the serial numbers (...0000 - ...9999) can be chosen with equal probability.