find the distance between the point 𝐒 and the line 𝔏 with the given equation. 𝐒(-2,4) ; 5 x-4 y-

step 1 Hello friends we have to find out the distance between the point is minus 2 comma 4 and the line

Find the distance between the point 𝐒 and the line 𝔏 with...

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Key Concepts

Distance from a Point to a Line

This concept involves calculating the shortest distance from a given point to a specified line in the plane. The formula |ax0 + by0 + c|/?(a² + b²) (where ax+by+c=0 is the line’s equation and (x0, y0) is the point) is derived by finding the length of the perpendicular dropped from the point to the line. It is a fundamental tool in analytic geometry for measuring separations in Euclidean spaces.

Standard Form of a Line Equation

Expressing a line in the form ax + by + c = 0 is crucial because it standardizes the representation, making it easier to apply various analytic and algebraic techniques. This form allows for a straightforward use of formulas involving distances, slopes, and intersections, and helps in computing the perpendicular distance from a point to the line.

Perpendicular Distance

The shortest distance from a point to a line is measured along the perpendicular from the point to the line. This concept is significant because it minimizes distance and is used in optimization problems, geometric constructions, and proofs. It ensures that the computed distance is the minimum possible, as any other line drawn from the point to the line would be longer.

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