x2+(y-3√2x)2=1

x2+(y-3√2x)2=1 Meaning and Solution of Mathematical Equation with Graph

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x2+(y-3√2x)2=1 Meaning: This equation represents a shape called an ellipse, but it looks a little different than the usual one. To understand it, we have to talk about shapes, math, and how they relate to each other.

Also Read: x2-11x+28=0

Understand about x2+(y-3√2x)2=1 Mathematical Equation

This equation is like a special rule that tells us about points on a circle. The x and y are like coordinates on a graph, and the equation helps us figure out which points are on the circle and which are not. It’s like a puzzle that helps us understand where things are in relation to each other.

  • Standard Form of an Ellipse: An ellipse is a shape that looks like a stretched out circle. The standard form equation for an ellipse is x^2/a^2 + y^2/b^2 = 1, where a and b are the lengths of the longer and shorter sides of the ellipse.
  • Transformation: Imagine a magical equation that tells us how things change. In this equation, there is a special number that makes things move and change size. It’s like a secret code that helps us understand how things shift and grow.
  • Geometric Interpretation: This equation is like a special shape called an ellipse that has been moved and turned from its normal position. The turning is because of the number in the x part of the equation, and the moving is because the center of the ellipse is different.

Solve x2+(y-3√2x)2=1 Mathematical Equation

To figure out how an ellipse works, we can change the equation to make it easier to understand.

Making the equation bigger and changing the order of the terms can help us understand it better. We can figure out how long and which way the ellipse is by using math tricks like completing the square.

  • We can figure out exactly where the ellipse is on a graph by finding the center point of the ellipse.
  • Breaking down the equation into smaller parts to understand it better.
  • To better understand this equation, let’s break down the term that is squared.

Complete Analysis of the Equation

Algebraic Expansion: When we have a number, y, minus three times the square root of two to the power of another number, x, and then we square that whole thing, it is equal to y squared minus six times the square root of two times y times x, plus eighteen times x squared ((y−3√2​^x)2=y^2−6√2​xy+18x^2).

Putting this back into the original equation will show the answer.

x^2+y^2−6√2​xy+18x^2=1

When you have a square and a circle, and you put them together in a certain way, you get the number 1.

When we change the order of the terms in an equation, it becomes a different shape that looks like a curved line or a circle.

There is an equation with numbers and letters in it. The equation is 19 times the number x squared minus 6 times the square root of 2 times the number x times the number y, plus the number y squared, minus 1, equals 0 (19x^2−6√2​xy+y^2−1=0)

Finding the Rotation: If there is an xy term in the equation, it means the ellipse is tilted. We can figure out how much it is tilted by using a special formula that helps us understand the shape of the ellipse better.

This is about using special math to figure out measurements for shapes and angles.

When we turn the shape around, we can make the equation simpler and see how long the oval’s two sides are. It’s important to draw and look at a picture of the oval to see how it looks different when it’s turned.

The center of an ellipse is like the middle point of it. To find the center, we need to know how the ellipse has been moved or shifted from its starting position.

Axis Orientation: Once we figure out how something is spinning, we can also figure out which way it is pointing, which is important for things that need to know which way they are going. Application and Visualization: This helps us use and see things in a clear and accurate way.

Application and Visualization

  • Graphing: Graphing means drawing a picture of a shape called an ellipse on a grid. You can use a computer program or do it by hand to find important points on the shape.

  • Real-World Applications: Ellipses are shapes that we can see in space, like the paths that planets take. They are also used by scientists and engineers to solve problems and understand how things work. Knowing how to write down the equations for ellipses is really important in these areas.
  • Analyzing Properties: Studying the shape of an ellipse can involve figuring out how squished or stretched out it is compared to a circle. We can also look at other things like how long certain lines are within the ellipse.

Conclusion

The conclusion is like the ending of a story or the final part of an experiment. It’s when we sum up everything we learned or discovered and make a final decision or statement about it. It’s like the last puzzle piece that helps us understand the whole picture.

The equation x2+(y-3√2x)2=1 is not just something interesting in math, it can also help us learn about ellipses and how they are used in different ways. To understand this equation, we need to use math rules, think about shapes, and imagine things in our mind. It is a really cool thing to learn about in the subject of math.

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