Proving Points On A Locus Understanding Coordinate Geometry

In the realm of coordinate geometry, the concept of a locus is fundamental. A locus is essentially the set of all points that satisfy a given condition or equation. Understanding how to prove that a point lies on a locus is a crucial skill in mathematics. This article delves into the methods of proving that specific points lie on a locus, using examples involving quadratic and linear equations. We will explore how to substitute coordinates into equations and verify their validity. This detailed exploration aims to enhance your understanding of coordinate geometry and problem-solving techniques.

In this section, we are given that the point (4, 4) lies on the locus defined by the equation y² = ax. Our task is twofold: first, determine the value of the constant 'a,' and second, prove that the point (16, 8) also lies on this locus. This involves substituting the coordinates of the given point into the equation to find 'a,' and then using this value to verify if the second point satisfies the same equation. This exercise not only reinforces the concept of a locus but also highlights the importance of algebraic manipulation and substitution in coordinate geometry.

Step 1: Finding the Value of 'a'

To begin, we know that the point (4, 4) lies on the locus y² = ax. This means that when we substitute x = 4 and y = 4 into the equation, it should hold true. Let's perform this substitution:

y² = ax

4² = a * 4

16 = 4a

Now, to find the value of 'a,' we divide both sides of the equation by 4:

a = 16 / 4

a = 4

Therefore, we have determined that the value of 'a' is 4. This constant defines the specific parabola in our case. Understanding how to derive such constants is vital in many mathematical contexts, as it allows us to define specific curves and relationships based on given conditions. The ability to isolate variables and solve for unknowns is a fundamental skill in algebra, and this step clearly demonstrates its application.

Step 2: Verifying (16, 8) Lies on the Locus

Now that we have found the value of 'a,' which is 4, we can rewrite the equation of the locus as y² = 4x. To prove that the point (16, 8) lies on this locus, we need to substitute x = 16 and y = 8 into the equation and check if the equation holds true. This process is a direct application of the definition of a locus: a point lies on a locus if and only if its coordinates satisfy the equation of the locus.

Substituting the coordinates of the point (16, 8) into the equation y² = 4x, we get:

8² = 4 * 16

64 = 64

Since the equation holds true, we have successfully demonstrated that the point (16, 8) indeed lies on the locus y² = 4x. This verification step is crucial in confirming our understanding of the locus and the correctness of our calculations. It showcases the power of substitution in verifying geometric properties using algebraic equations.

In this part, we are given that the point (2, 3) lies on the locus defined by the equation ax + 2y = 16. Our objective is to first determine the value of 'a' and then demonstrate that the point (0, 8) also lies on this locus. This task is similar to the previous one but involves a linear equation instead of a quadratic one. This provides a different context for applying the same fundamental principles of coordinate geometry.

Step 1: Determining the Value of 'a'

Given that the point (2, 3) lies on the locus ax + 2y = 16, we can substitute x = 2 and y = 3 into the equation to find the value of 'a.' This substitution allows us to create an equation with 'a' as the only unknown, which we can then solve using algebraic techniques. This process highlights the versatility of algebraic methods in solving geometric problems.

Substituting the coordinates, we get:

a * 2 + 2 * 3 = 16

2a + 6 = 16

Now, we subtract 6 from both sides of the equation:

2a = 16 - 6

2a = 10

Finally, we divide both sides by 2 to solve for 'a':

a = 10 / 2

a = 5

Thus, we have found that the value of 'a' is 5. This value is specific to the line defined by the given conditions. The ability to determine coefficients in linear equations is essential for understanding and manipulating linear relationships in various fields, including mathematics, physics, and engineering.

Step 2: Demonstrating (0, 8) Lies on the Locus

Now that we know a = 5, the equation of the locus is 5x + 2y = 16. To show that the point (0, 8) lies on this locus, we substitute x = 0 and y = 8 into the equation and check if it holds true. This step is a direct application of the definition of a locus, ensuring that the coordinates of the point satisfy the equation.

Substituting the coordinates of the point (0, 8) into the equation 5x + 2y = 16, we have:

5 * 0 + 2 * 8 = 16

0 + 16 = 16

16 = 16

Since the equation is true, we have successfully shown that the point (0, 8) lies on the locus 5x + 2y = 16. This demonstrates the power of using algebraic equations to verify geometric properties. The ability to confirm whether a point lies on a line or curve is a fundamental skill in coordinate geometry and has numerous applications in various mathematical and scientific disciplines.

In this article, we have explored the process of proving that points lie on a locus, using both quadratic and linear equations. We have demonstrated how to determine unknown constants within the equations and how to verify if a point satisfies the equation of a locus. These skills are essential in coordinate geometry and provide a foundation for more advanced mathematical concepts. By mastering these techniques, you can confidently tackle problems involving loci and their equations. The ability to substitute values, solve for unknowns, and verify solutions are crucial skills not only in mathematics but also in various scientific and engineering fields. Understanding the concept of a locus and how to work with it provides a valuable tool for problem-solving and analytical thinking.