Normal Line Equation: Finding K Value

Oct 10, 2025 by ADMIN 38 views

Persamaan Garis Normal terhadap Kurva: Mencari Nilai k

Hey guys! Let's dive into a fun math problem involving curves, normal lines, and finding the value of k. This is a classic calculus problem that combines concepts from coordinate geometry and differential calculus. We're given a curve, a point on that curve, and a condition about the normal line at that point. Our mission? Find the value of k that makes everything tie together. Buckle up, it's gonna be a mathematical ride!

Memahami Soal

So, the question states that we have a curve defined by the equation $y = 2x^2 + kx + 1$ . We also know that the point (-1, 2) lies on this curve. Furthermore, the normal line to the curve at this point is parallel to the line $3y - x = 9$ . Our goal is to determine the value of k. To solve this, we'll need to understand the concepts of derivatives, normal lines, and parallel lines. First, the derivative of a curve at a point gives us the slope of the tangent line at that point. The normal line is perpendicular to the tangent line, so its slope is the negative reciprocal of the tangent line's slope. Parallel lines have the same slope. By using these concepts, we can set up equations and solve for k.

Langkah-langkah Penyelesaian

Okay, let's break this down step by step. First, ensure that the point (-1, 2) actually lies on the curve. Substitute x = -1 and y = 2 into the equation of the curve: $2 = 2(-1)^2 + k(-1) + 1$ . Simplifying this, we get $2 = 2 - k + 1$ , which further simplifies to $2 = 3 - k$ . Solving for k, we find $k = 1$ . However, this only ensures that the point lies on the curve. We still need to consider the condition about the normal line. Next, find the derivative of the curve $y = 2x^2 + kx + 1$ . The derivative, dy/dx, represents the slope of the tangent line. Using the power rule, we find that $dy/dx = 4x + k$ . Now, evaluate the derivative at the point x = -1. This gives us the slope of the tangent line at that point: $dy/dx |_{-1} = 4(-1) + k = -4 + k$ . The slope of the normal line is the negative reciprocal of the tangent line's slope. Therefore, the slope of the normal line is $-1/(-4 + k) = 1/(4 - k)$ .

Mencari Gradien Garis yang Sejajar

Now, let's find the slope of the line $3y - x = 9$ . To do this, we can rewrite the equation in slope-intercept form (y = mx + b), where m is the slope. Adding x to both sides gives $3y = x + 9$ . Dividing by 3, we get $y = (1/3)x + 3$ . Thus, the slope of this line is 1/3. Since the normal line is parallel to this line, their slopes must be equal. Therefore, we have $1/(4 - k) = 1/3$ . To solve for k, we can cross-multiply: $3 = 4 - k$ . Solving for k, we get $k = 4 - 3 = 1$ . So, the value of k that satisfies all the given conditions is 1.

Pembahasan Mendalam

The key to solving this problem lies in understanding the relationship between the curve, its tangent, and its normal line. The derivative of the curve provides the slope of the tangent line at a specific point. The normal line, being perpendicular to the tangent, has a slope that is the negative reciprocal of the tangent's slope. Furthermore, the condition that the normal line is parallel to another given line means that their slopes are equal. By equating the expressions for these slopes and solving for k, we can find the value that satisfies all the conditions.

Konsep Turunan

Understanding derivatives is crucial. The derivative of a function, often denoted as dy/dx, represents the instantaneous rate of change of the function with respect to its variable. Geometrically, it gives the slope of the tangent line to the curve at a particular point. In this problem, we used the power rule to find the derivative of the curve $y = 2x^2 + kx + 1$ . The power rule states that if $y = ax^n$ , then $dy/dx = nax^(n-1)$ . Applying this rule to each term in the equation, we found that $dy/dx = 4x + k$ . This expression allows us to calculate the slope of the tangent line at any point on the curve by substituting the x-coordinate of that point.

Garis Normal dan Garis Singgung

The tangent line and normal line are perpendicular to each other. If the slope of the tangent line at a point is m, then the slope of the normal line at that point is -1/m. This relationship is fundamental in solving problems involving normal lines. In this case, we found the slope of the tangent line at x = -1 to be -4 + k. Therefore, the slope of the normal line is $1/(4 - k)$ . This expression is crucial because it allows us to relate the slope of the normal line to the unknown value of k.

Garis Sejajar

Parallel lines have the same slope. This is a fundamental concept in coordinate geometry. If two lines are parallel, their slopes are equal. In this problem, we were given that the normal line is parallel to the line $3y - x = 9$ . By rewriting the equation of this line in slope-intercept form, we found its slope to be 1/3. Since the normal line is parallel to this line, its slope must also be 1/3. This gives us the equation $1/(4 - k) = 1/3$ , which we can solve for k.

Jawaban yang Benar

From our calculations, we found that the value of k that satisfies all the given conditions is 1. Therefore, the correct answer is C. 1.

Tips and Tricks

Always double-check your calculations. A small arithmetic error can lead to a wrong answer. It's a good idea to review each step of your solution to ensure accuracy.
Understand the underlying concepts. Don't just memorize formulas. Make sure you understand the concepts of derivatives, tangent lines, normal lines, and parallel lines.
Practice, practice, practice. The more problems you solve, the better you'll become at recognizing patterns and applying the appropriate techniques.
Draw a diagram. Visualizing the problem can often help you understand the relationships between the different elements.

Kesimpulan

Alright, that was a fun one! We successfully navigated through the world of curves, tangents, and normals to find the value of k. Remember, understanding the core concepts is key to tackling these types of problems. Keep practicing, and you'll be a calculus whiz in no time! Keep your math skills sharp, and you'll be able to solve any problem that comes your way.