Finding Point P And Line NP: A Step-by-Step Guide
Hey everyone! Let's dive into a cool math problem involving coordinate geometry. We're going to find the coordinates of a point that divides a line segment internally in a given ratio and then determine the equation of a line passing through that point and another given point. Sounds fun, right? Let's get started!
Problem Statement
So, here's the deal: We have a straight line formed by joining two points, X(1, -2) and Y(5, 3). A point P divides this line internally in the ratio 2:3. Our mission, should we choose to accept it, is twofold:
(a) Find the coordinates of point P. (b) Determine the equation of the straight line that passes through N(3, -5) and P.
(a) Finding the Coordinates of Point P
The Section Formula: Our Trusty Tool
To find the coordinates of point P, we'll use the section formula. This formula is a gem when dealing with points dividing a line segment in a given ratio. It states that if a point P(x, y) divides the line segment joining A(x₁, y₁) and B(x₂, y₂) internally in the ratio m:n, then the coordinates of P are given by:
x = (m * x₂ + n * x₁) / (m + n) y = (m * y₂ + n * y₁) / (m + n)
Applying the Section Formula
In our case, we have:
X(x₁, y₁) = (1, -2) Y(x₂, y₂) = (5, 3) Ratio m:n = 2:3
Let's plug these values into the section formula to find the coordinates of P(x, y). First, we'll calculate the x-coordinate:
x = (2 * 5 + 3 * 1) / (2 + 3) x = (10 + 3) / 5 x = 13 / 5
Now, let's find the y-coordinate:
y = (2 * 3 + 3 * (-2)) / (2 + 3) y = (6 - 6) / 5 y = 0 / 5 y = 0
So, there you have it! The coordinates of point P are (13/5, 0). We've successfully located our point P, which divides the line segment XY in the ratio 2:3. This wasn't so bad, was it? The section formula really comes in handy for these types of problems, and it's a great tool to have in your math arsenal. Remember, the key is to correctly identify your x₁, y₁, x₂, y₂, and the ratio m:n, then just plug the values into the formula and do the math. Easy peasy!
Remember: Always double-check your calculations to avoid any silly mistakes. Math can be fun, especially when you get the right answer! Now, let's move on to the next part of our adventure: finding the equation of the straight line that passes through N and P.
(b) Finding the Equation of the Straight Line Through N and P
The Two-Point Form: Our Next Superpower
Now that we've found the coordinates of point P (13/5, 0), we need to determine the equation of the straight line that passes through P and another point, N(3, -5). For this, we'll use the two-point form of a line equation. This form is perfect when we know two points on the line. The two-point form states that the equation of a line passing through points (x₁, y₁) and (x₂, y₂) is given by:
(y - y₁) / (x - x₁) = (y₂ - y₁) / (x₂ - x₁)
Applying the Two-Point Form
In our case, we have:
P(x₁, y₁) = (13/5, 0) N(x₂, y₂) = (3, -5)
Let's plug these values into the two-point form to find the equation of the line. We'll start by substituting the coordinates into the formula:
(y - 0) / (x - 13/5) = (-5 - 0) / (3 - 13/5)
Now, let's simplify this equation step by step. First, we'll simplify the fractions:
y / (x - 13/5) = -5 / (15/5 - 13/5) y / (x - 13/5) = -5 / (2/5)
To get rid of the fraction in the denominator on the right side, we can multiply -5 by the reciprocal of 2/5, which is 5/2:
y / (x - 13/5) = -5 * (5/2) y / (x - 13/5) = -25/2
Next, let's get rid of the fraction in the denominator on the left side. We'll multiply both sides of the equation by (x - 13/5):
y = (-25/2) * (x - 13/5)
Now, let's distribute the -25/2 on the right side:
y = (-25/2)x + (25/2) * (13/5)
Simplify the second term:
y = (-25/2)x + (5 * 13) / 2 y = (-25/2)x + 65/2
To get rid of the fractions, we can multiply the entire equation by 2:
2y = -25x + 65
Finally, let's rearrange the equation into the standard form of a linear equation, which is Ax + By = C:
25x + 2y = 65
And there you have it! The equation of the straight line that passes through N(3, -5) and P(13/5, 0) is 25x + 2y = 65. We did it! We successfully navigated through the steps, applying the two-point form, simplifying fractions, and rearranging the equation. This is a great example of how different mathematical tools can come together to solve a problem. The key here is to stay organized, take it one step at a time, and don't be afraid of fractions – they're just numbers too!
Pro Tip: Always double-check your work by plugging the coordinates of points N and P back into the equation to make sure they satisfy it. This is a great way to catch any errors and ensure that your answer is correct.
Conclusion
So, guys, we've tackled a pretty neat problem today! We found the coordinates of point P using the section formula and then determined the equation of the line passing through N and P using the two-point form. These are fundamental concepts in coordinate geometry, and mastering them will definitely help you in your mathematical journey. Remember, practice makes perfect, so keep solving problems and exploring new concepts. Math can be challenging, but it's also incredibly rewarding when you crack a tough problem. Keep up the great work, and happy problem-solving!
