How to Determine the Equation of a Straight Line with Given Angle and Intercept
Understanding and determining the equation of a straight line based on given angles and intercepts is a fundamental concept in analytic geometry. In this article, we will delve into how to derive the equation of a straight line that forms a specific angle with the x-axis and has a certain intercept on the negative y-axis. We will use the equation format y mx c where m represents the slope and c the y-intercept.
Deriving the Equation from Given Parameters
Given that the line makes an angle with the x-axis such that the tangent of this angle is 2, and it cuts off an intercept of -5 from the negative y-axis, our task is to find the equation of this straight line. Let's break it down step by step:
Determine the slope: The slope m of the line is given by the tangent of the angle it makes with the x-axis. Since it is given that tan(θ) 2, the slope m 2. Determine the y-intercept: The y-intercept c is the point where the line intersects the y-axis. Since the problem specifies that this intercept is -5 and it is on the negative y-axis, we have c -5. Formulate the equation: Using the slope-intercept form of the equation y mx c, substitute the values of m and c. Therefore, the equation of the line is:y 2x - 5
Explanation and Illustration
The equation y 2x - 5 ensures that the line has a slope of 2 and a y-intercept of -5. This means that for every unit increase in x, y increases by 2 units, and the line crosses the y-axis at (0, -5).
Below is a graphical representation of the line y 2x - 5 for reference:
Conclusion
Understanding how to derive the equation of a straight line using the given angle and intercept is crucial in various applications, including physics, engineering, and mathematics. By following the steps outlined, you can easily find the equation of any straight line based on the provided conditions.
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