In Exercises 45–52, use your answers from Exercises 41–44 and the parametric equations given in Exercises 41–44 to find a set of parametric equations for the conic section or the line.

Line: Passes through (−2,4) and (1,7)

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Identify the two points given: \((-2, 4)\) and \((1, 7)\).

Calculate the slope \(m\) of the line using the formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\).

Substitute the coordinates into the slope formula: \(m = \frac{7 - 4}{1 - (-2)}\).

Use the point-slope form of a line to write the equation: \(y - y_1 = m(x - x_1)\), choosing one of the points, for example \((-2, 4)\).

Express the parametric equations by letting \(x = t\) (a parameter), then find \(y\) in terms of \(t\) using the line equation: \(y = m(t - x_1) + y_1\).

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Parametric Equations of a Line

Parametric equations express the coordinates of points on a line as functions of a parameter, usually t. For a line through points (x₁, y₁) and (x₂, y₂), the equations are x = x₁ + t(x₂ - x₁) and y = y₁ + t(y₂ - y₁), describing all points between and beyond these points.

Vector Representation of a Line

A line can be represented using a position vector and a direction vector. The direction vector is found by subtracting the coordinates of the two given points, indicating the line's slope and direction, which is essential for forming parametric equations.

Using Given Points to Determine Parameters

Given two points, you can determine the parametric form by setting one point as the initial position and using the difference between points as the direction. This approach ensures the parametric equations accurately represent the line passing through both points.

In Exercises 45–52, use your answers from Exercises 41–44 and the parametric equations given in Exercises 41–44 to find a set of parametric equations for the conic section or the line.Line: Passes through (−2,4) and (1,7)