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Tuesday, December 31, 2013

Find the relation between x and y, if the point (x, y) is equidistant from (7, -6),(-3, 4)

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How to find the relation of x and y.

Problem: How to Find the relation between x and y, if the point (x, y)is equidistant from (7, -6),(-3, 4).

(7,-6),(-3,4)\,

Solutions:

Find the distance between (x,y) and the first point is

\sqrt{(x-7)^2+(y+6)^2}\,

\sqrt{x^2+49-14x+y^2+36+12y}\,

\sqrt{x^2+y^2-14x+12y+85}\,

Let this expression be 1.

Find the distance between (x,y) and the second point is

\sqrt{(x+3)^2+(y-4)^2}\,

\sqrt{x^2+9+6x+y^2+16-8y}\,

\sqrt{x^2+y^2+6x-8y+25}\,

Let this expression be 2.

As per the given condition,equating both the expressions,

\sqrt{x^2+y^2-14x-12y+85}=\sqrt{x^2+y^2+6x-8y+25}\,

Squaring on both sides,we get

x^2+y^2-14x+12y+22=x^2+y^2+6x-8y+25\,

cancelling the terms which are common on both sides,

-14x-6x+12y+8y=-60\,

-20x+20y=-60\,

List of Similar Problems with Complete Solutions

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credit: Todd©2013 www.FroydWess.com

Find the value of a if the distance between the points (a, 2),(3, 4) is 2

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Value of a if the distance between the points (a, 2),(3, 4) is 2

Problem: How to find the value of a if the distance between the points (a, 2),(3, 4) is 2.

(a,2),(3,4)\,

Solutions:

The distance between the given points is

\sqrt{(a-3)^2+(2-4)^2}=2\,

Simplifying

\sqrt{a^2+9-6a+4}=2\,

Squaring on both sides,

a^2-6a+13=4\,

a^2-6a+9=0\,

(a-3)^2=0\,

Finally, the value of a is 3.

List of Similar Problems with Complete Solutions

NOTE:

For any Questions, Clarifications and any violent Reactions use the comment section below. I'll be much willing to assist you.


credit: Todd©2013 www.FroydWess.com

10 Most Amazing Inventions Of 2013

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technology_2013

With advancements in technology making greater leaps and bounds every year, 2013 was filled with some of the greatest creations to ever be imagined by mankind. Here are 10 such amazing technological feats that stood out.

1. 3Doodler

3doodler

3-D printing has been around for quite some time now, but the 3Doodler is the world’s first 3-D printing pen. This unique gadget makes 3-D printing portable. Created by Peter Dilworth and Maxwell Bogue of WobbleWorks LLC, the 3Doodler uses Poly-Lactic Acid (acrylonitrile butadiene styrene) to create three-dimensional pictures. The pen melts the plastic (available in different colors) and once it is cooled it forms a solid 3-D structure.

2. Mission R

missionr_electric_bike

Electric cars are becoming more and more common every day, with automotive giants producing their own version of the product. The Mission R, however, puts the concept of an electric bike into perspective. With the ability to do 0-60 mph in three seconds and a top speed of 150 mph, this bike can certainly compete with any sports bike out there. Customers can choose from a variety of battery packs depending on how they want to use the bike. The battery packs range from a 12kWh battery (range: 105 miles) to 17kWh battery (range: 140 miles).

3. Volvo Solar Pavilion

volvo_solar_pavilion

Electric cars are great for the environment but sadly they have a limited range and it’s difficult to find a place to charge them in the middle of a journey. The Volvo Solar Pavilion is a flexible pavilion that has numerous solar panels on its surface. The pavilion connects to the car just like any power outlet and uses solar power to charge the car’s battery. Once the vehicle is charged, the pavilion can be folded up and stored in the boot of the car. The Volvo Solar Pavilion is a great solution to the problem of limited range which every electric vehicle faces.

4. Sony Smart Lens

sony_smart_lens

For those of you who want more from your smartphone’s camera or for a professional photographer looking for a more compact solution for getting the best quality pictures, the Sony Smart Lens is the perfect gadget. The lens connects with your smartphone via NFC or built-in WiFi and is controlled by the respective app available for both iPhone and Android. The app allows you to use your phone as a viewfinder for the lens and also to customize settings. The lens and phone can work together even when not attached.

5. Oculus Rift

oculus_rift

With new gaming systems being unveiled in 2013, the Oculus Rift could not have come at a better time. Developed by Oculus VR, the Rift is a head mounted virtual reality display which is still in the development phase. The gadget weighs only 379 grams and can be connected via HDMI, DVI and USB port. The Rift includes a 1920×1080 7-inch 3-D display and many game developers are giving support for it.

6. GravityLight

gravity_light

Made by Deciwatt, the GravityLight is a device which uses the force of gravity to create light. It was created with the people of third world countries in mind, who do not have access to electricity. A weight is attached to the device and pulled to a certain height, which powers and internal generator as it slowly falls down and creates enough power to light up the LEDs in the device. The light lasts 25 minutes and it only takes 3 seconds to set-up the weight.

7. Atlas Robot

darpa_atlas

This humanoid robot created by Boston Dynamics, consists of four limbs and two vision systems (one of which uses stereo cameras while the other is a laser range finder). Atlas is made from aluminium and titanium of the same quality as that used in the manufacturing of aircrafts. Standing at 6 feet tall and weighing 150 kg, this robot is meant for search and rescue missions in situations that are too dangerous for humans.

8. Invisible Skyscraper

invisible_skyscraper

Planned to be built in South Korea, Tower Infinity will be the world’s first invisible skyscraper. The skyscraper uses cameras to capture images of the surrounding scenery and projects them onto the building using LED projectors which make it invisible to people. The 450 meter tall invisible building was designed by GDS Architects and although it will be invisible to pedestrians, the structure will still be visible to planes and birds.

9. Makerbot Digitizer

makerbot_digitizer

This 3-D scanner lets you scan anything that can fit on to it, edit the digital image and even print a 3-D version of the scanned object. The device has a turntable on which the object is placed, a few lasers and a 1.3 megapixel CMOS sensor for precise scanning of the objects. The software that comes with the scanner allow you to edit the scanned images and print them using a 3-D printer.

10. Google Glass

google_glass

Google Glass is probably the most notable tech achievement of 2013. With this gadget, sci-fi has become reality and users can take pictures, make videos and access the internet without even having to reach into their pockets. This wearable technology is a big leap forward. Glass has a 5 megapixel camera and can record 720p videos. The device is operated using voice command and is basically a wearable computer.


credit: Uber Geek of wonderfulengineering.com©2013 www.FroydWess.com

Monday, December 30, 2013

List of Finding Equation of a Line Problems with Complete Solutions

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List of Finding Equation of a Line Problems with Complete Solutions
This page contains the complete list of problems with solutions in how to find equation of the line. The basic and one of the interesting topics in Analytic Geometry. These examples will definitely make it easy for you to learn and solve Geometry problems. Each problems is carefully chosen as a guide to solve similar problems in your seatwork, homework, quizzes, exercises and Board exams.

Equation of a Line Problems in Geometry with Complete Solutions


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For any Questions, Clarifications and any violent Reactions use the comment section below. I'll be much willing to assist you.


credit: ©2013 www.FroydWess.com

List of Finding Slope Problems with Complete Solutions

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List of Finding Slope Problems with Complete Solutions
This page contains the complete list of problems with solutions in how to find Slope of the line. The basic and one of the interesting topics in Analytic Geometry. These examples will definitely make it easy for you to learn and solve Geometry problems. Each problems is carefully chosen as a guide to solve similar problems in your seatwork, homework, quizzes, exercises and Board exams.

Slope Problems in Geometry with Complete Solutions


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For any Questions, Clarifications and any violent Reactions use the comment section below. I'll be much willing to assist you.


credit: ©2013 www.FroydWess.com

List of Finding Midpoint Problems with Complete Solutions

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List of Finding Midpoint Problems with Complete Solutions
This page contains the complete list of problems with solutions in how to find midpoint of the line. The basic and one of the interesting topics in Analytic Geometry. These examples will definitely make it easy for you to learn and solve Geometry problems. Each problems is carefully chosen as a guide to solve similar problems in your seatwork, homework, quizzes, exercises and Board exams.

Midpoint Problems in Geometry with Complete Solutions


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For any Questions, Clarifications and any violent Reactions use the comment section below. I'll be much willing to assist you.


credit: ©2013 www.FroydWess.com

List of Distance Problems with Complete Solutions

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List of Distance Problems with Complete Solutions
This page contains the complete list of problems with solutions in how to find distance between two points. The basic and one of the interesting topics in Analytic Geometry. These examples will definitely make it easy for you to learn and solve Geometry problems. Each problems is carefully chosen as a guide to solve similar problems in your seatwork, homework, quizzes, exercises and Board exams.

Distance Problems in Geometry with Complete Solutions


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For any Questions, Clarifications and any violent Reactions use the comment section below. I'll be much willing to assist you.


credit: ©2013 www.FroydWess.com

List of Geometry Problems with Complete Solutions

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Geometry Problems with Complete Solutions
This page contains the complete list of problems in Geometry with solutions. These examples will definitely make it easy for you to learn and solve Geometry problems. Each problems is carefully chosen as a guide to solve similar problems in your seatwork, homework, quizzes, exercises and Board exams.

Geometry Problems with Complete Solutions

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For any Questions, Clarifications and any violent Reactions use the comment section below. I'll be much willing to assist you.


credit: ©2013 www.FroydWess.com

Saturday, December 28, 2013

Given (0, 9) and (6, -9), find the midpoint, distance, slope, and equation of the line.

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Problem: Given these pairs of points, (0, 9)) and (6, -9), find the midpoint, distance, slope, and equation of the line.

(0,9),(6,-9)\,

Solutions:

  • To find the midpoint, average the x coordinates and y coordinates. The midpoint is

\left(\frac{0+6}{2},\frac{9-9}{2}\right) = (3,0)\,

  • To find the (always zero or positive) distance, use the formula

d = +\sqrt{(x_1-x_2)^2 + (y_1-y_2)^2}\,

d = \sqrt{(6)^2+(-9-9)^2} = \sqrt{36+18^2} = \sqrt{(2\cdot 3)^2+(2\cdot 3^2)^2} = \sqrt{(2\cdot 3)^2(1+3^2} = 6\sqrt{10}\,

d = \sqrt{(6)^2+(-9-9)^2} = \sqrt{36+18^2} = \sqrt{(2\cdot 3)^2+(2\cdot 3^2)^2} = \sqrt{(2\cdot 3)^2(1+3^2} = 6\sqrt{10}\,

  • To find the slope, use the formula

 m = \frac{y_2-y_1}{x_2-x_1}\,

m = \frac{-9-9}{6-0} = -3\,

The equations of the line are

Method 1:

 y=mx+b\,

  • Plug in one known point (say, (0, 9)) and the calculated slope.

9 = -3\cdot 0 + b\,

b = 9\,

  • Now plug b and m into the line equation:
  • y = -3x + 9\,

Method 2:

(y-y_1) = m(x-x_1)\,

  • Plug in one known point (say, (6, –9) ) and the calculated slope.

(y+9) = -3(x-6)\,

y = -3x + 18 - 9\,

  • y = -3x + 9\,

List of Similar Problems with Complete Solutions

NOTE:

For any Questions, Clarifications and any violent Reactions use the comment section below. I'll be much willing to assist you.


credit: Todd©2013 www.FroydWess.com
 

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