The Co-ordinate Of A Point On YX Is Equidistant From The Points 6,5 And -4 3​

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Introduction

In coordinate geometry, the concept of equidistance plays a crucial role in determining the position of a point with respect to other points. The distance between two points in a coordinate plane is calculated using the distance formula, which is derived from the Pythagorean theorem. In this article, we will explore the concept of equidistance and use it to find the co-ordinate of a point on YX that is equidistant from the points 6,5 and -4,3.

What is Equidistance?

Equidistance refers to the state of being equally distant from two or more points. In other words, a point is said to be equidistant from two points if the distance between the point and each of the two points is the same. This concept is widely used in geometry, trigonometry, and other branches of mathematics.

The Distance Formula

The distance formula is used to calculate the distance between two points in a coordinate plane. The formula is given by:

d = √((x2 - x1)² + (y2 - y1)²)

where (x1, y1) and (x2, y2) are the co-ordinates of the two points.

Finding the Co-ordinate of a Point on YX

To find the co-ordinate of a point on YX that is equidistant from the points 6,5 and -4,3, we need to use the distance formula. Let the co-ordinate of the point be (x, y). Then, the distance between the point and the point 6,5 is given by:

d1 = √((x - 6)² + (y - 5)²)

Similarly, the distance between the point and the point -4,3 is given by:

d2 = √((x + 4)² + (y - 3)²)

Since the point is equidistant from the two points, we can set up the equation:

d1 = d2

Substituting the values of d1 and d2, we get:

√((x - 6)² + (y - 5)²) = √((x + 4)² + (y - 3)²)

Squaring both sides of the equation, we get:

(x - 6)² + (y - 5)² = (x + 4)² + (y - 3)²

Expanding the squares, we get:

x² - 12x + 36 + y² - 10y + 25 = x² + 8x + 16 + y² - 6y + 9

Simplifying the equation, we get:

-12x - 10y + 61 = 8x - 6y + 25

Combine like terms:

-20x - 4y + 36 = 0

Solving the Equation

To solve the equation, we need to isolate the variables x and y. We can do this by rearranging the equation to get:

-20x - 4y = -36

Divide both sides of the equation by -4:

5x + y = 9

Finding the Co-ordinate of the Point

Now that we have the equation 5x + y = 9, we can find the co-ordinate of the point by substituting the values of x and y. However, we need to find the values of x and y that satisfy the equation.

To do this, we can use the method of substitution or elimination. Let's use the method of substitution. We can substitute the value of y from the equation 5x + y = 9 into the equation x² - 12x + 36 + y² - 10y + 25 = x² + 8x + 16 + y² - 6y + 9.

Substituting the value of y, we get:

x² - 12x + 36 + (9 - 5x)² - 10(9 - 5x) + 25 = x² + 8x + 16 + (9 - 5x)² - 6(9 - 5x) + 9

Simplifying the equation, we get:

x² - 12x + 36 + 81 - 90x + 25x² + 450 - 50x + 25 = x² + 8x + 16 + 81 - 90x + 25x² - 54 + 30x + 9

Combine like terms:

26x² - 62x + 496 = 26x² - 46x + 52

Subtract 26x² from both sides of the equation:

-62x + 496 = -46x + 52

Add 46x to both sides of the equation:

-16x + 496 = 52

Subtract 496 from both sides of the equation:

-16x = -444

Divide both sides of the equation by -16:

x = 27.75

Finding the Value of y

Now that we have the value of x, we can find the value of y by substituting the value of x into the equation 5x + y = 9.

Substituting the value of x, we get:

5(27.75) + y = 9

Multiply 5 by 27.75:

137.75 + y = 9

Subtract 137.75 from both sides of the equation:

y = -128.75

Conclusion

In this article, we used the concept of equidistance to find the co-ordinate of a point on YX that is equidistant from the points 6,5 and -4,3. We used the distance formula to calculate the distance between the point and each of the two points, and then set up an equation to find the co-ordinate of the point. We solved the equation using the method of substitution and found the values of x and y that satisfy the equation. The co-ordinate of the point is (27.75, -128.75).

Frequently Asked Questions

  • What is equidistance? Equidistance refers to the state of being equally distant from two or more points.
  • How do you calculate the distance between two points in a coordinate plane? The distance formula is used to calculate the distance between two points in a coordinate plane. The formula is given by: d = √((x2 - x1)² + (y2 - y1)²)
  • How do you find the co-ordinate of a point that is equidistant from two points? To find the co-ordinate of a point that is equidistant from two points, you need to use the distance formula and set up an equation to find the co-ordinate of the point.

Introduction

In our previous article, we explored the concept of equidistance and used it to find the co-ordinate of a point on YX that is equidistant from the points 6,5 and -4,3. In this article, we will answer some frequently asked questions related to the topic.

Q&A

Q: What is equidistance?

A: Equidistance refers to the state of being equally distant from two or more points.

Q: How do you calculate the distance between two points in a coordinate plane?

A: The distance formula is used to calculate the distance between two points in a coordinate plane. The formula is given by: d = √((x2 - x1)² + (y2 - y1)²)

Q: How do you find the co-ordinate of a point that is equidistant from two points?

A: To find the co-ordinate of a point that is equidistant from two points, you need to use the distance formula and set up an equation to find the co-ordinate of the point.

Q: What is the co-ordinate of a point on YX that is equidistant from the points 6,5 and -4,3?

A: The co-ordinate of the point is (27.75, -128.75).

Q: How do you solve the equation 5x + y = 9?

A: To solve the equation 5x + y = 9, you can use the method of substitution or elimination. Let's use the method of substitution. We can substitute the value of y from the equation 5x + y = 9 into the equation x² - 12x + 36 + y² - 10y + 25 = x² + 8x + 16 + y² - 6y + 9.

Q: What is the significance of the co-ordinate of a point on YX that is equidistant from two points?

A: The co-ordinate of a point on YX that is equidistant from two points is significant because it represents a point that is equally distant from two other points. This concept is widely used in geometry, trigonometry, and other branches of mathematics.

Q: How do you use the concept of equidistance in real-life applications?

A: The concept of equidistance is used in various real-life applications, such as:

  • Navigation: Equidistance is used to determine the shortest distance between two points on a map.
  • Architecture: Equidistance is used to design buildings and structures that are equally distant from each other.
  • Engineering: Equidistance is used to design systems and mechanisms that are equally distant from each other.

Conclusion

In this article, we answered some frequently asked questions related to the concept of equidistance and its application in finding the co-ordinate of a point on YX that is equidistant from the points 6,5 and -4,3. We hope that this article has provided you with a better understanding of the concept and its significance.

Frequently Asked Questions

  • What is the co-ordinate of a point on YX that is equidistant from the points 6,5 and -4,3? The co-ordinate of the point is (27.75, -128.75).
  • How do you solve the equation 5x + y = 9? To solve the equation 5x + y = 9, you can use the method of substitution or elimination.
  • What is the significance of the co-ordinate of a point on YX that is equidistant from two points? The co-ordinate of a point on YX that is equidistant from two points is significant because it represents a point that is equally distant from two other points.
  • How do you use the concept of equidistance in real-life applications? The concept of equidistance is used in various real-life applications, such as navigation, architecture, and engineering.