Finding the Equation of a Straight Line with Specific Slope and Angle Conditions
Finding the Equation of a Straight Line with Specific Slope and Angle
Finding the Equation of a Straight Line with Specific Slope and Angle Conditions
Determining the equation of a straight line when given specific conditions like slope and angle can be an interesting and mathematically challenging problem. This article provides a detailed solution to finding such a line that passes through a given point and makes a specific angle with another line.Introduction
In mathematics, the equation of a straight line is crucial in various applications, from geometry to physics. In this article, we will explore how to find the equation of a line that not only passes through a specific point but also makes a specific angle with another line. We will use the given conditions and employ basic trigonometry and algebra to solve this problem.Problem Statement
We are given points and a line with which the required line makes an angle such that the tangent of that angle is 2. The problem can be summarized as follows: The required line passes through the point (-10, 4). The given line is x - 2y 10. The tangent of the angle between the two lines is 2.Solving the Problem
To solve this problem, we need to follow a series of steps involving slope calculation, angle of inclination, and algebra.Step 1: Determine the slope of the given line.
The given line is x - 2y 10. We can convert this equation into the slope-intercept form (y mx b) to determine its slope.y 1 - 2 y 2 Which simplifies to: y 1 / 2 2 - x / 2 The slope, (m2 frac{1}{2}).
Step 2: Determine the slope of the required line.
We are given that the tangent of the angle between the two lines is 2. The formula for the tangent of the angle between two lines with slopes (m1) and (m2) is: tan ( θ ) m 1 - m 2 1 m 1 m 2 Substituting the given slope of the given line, (m2 frac{1}{2}), and the tangent of the angle, we get: m 1 - 1 2 1 m 1 · 1 2 2 This simplifies to a quadratic equation, which we solve to find (m1).Step 3: Solve the quadratic equation.
Rearranging the given equation, we have: m 1 - 1 2 1 m 1 2 2 Squaring both sides and simplifying, we get a quadratic equation in terms of (m1). Solving this equation will give us the possible values of (m1). We find that (m1 -frac{3}{4}) is the valid solution.Step 4: Find the equation of the required line.
The equation of the line with slope (m1 -frac{3}{4}) passing through point (-10, 4) can be found using the point-slope form of the line equation, (y - y1 m(x - x1)). y - 4 - 3 4 ( x - - 10 ) Simplifying this, we get: y - 4 - 3 4 ( x 10 ) - 15 4 Converting this into the general form (Ax By C 0), we get: 3 x 4 y 14 0 Thus, the equation of the required line is (3x 4y 14 0).