The Perpendicular Distance from a Point to a Line: d=|(A x1+B y1+C)/(√(A^2+B^2))| The perpendicu

step 1 In this question the point given is 1 comma by and equation of line is given which is y equals t

The Perpendicular Distance from a Point to a Line: d=|(A...

The Perpendicular Distance from a Point to a Line: $d=\left|\frac{A x_{1}+B y_{1}+C}{\sqrt{A^{2}+B^{2}}}\right|$ The perpendicular distance from a point $\left(x_{1}, y_{1}\right)$ to a given line can be found using the formula shown, where $A x+B y+C=0$ is the equation of the line in standard form $(A, B, \text { and } C$ are integers). Find the value(s) for $y$ that ensure $(1, y)$ is this same distance from $y=-\frac{1}{2} x+3$

The Perpendicular Distance from a Point to a Line: $d=\left|\frac{A x_{1}+B y_{1}+C}{\sqrt{A^{2}+B^{2}}}\right|$ The perpendicular distance from a point $\left(x_{1}, y_{1}\right)$ to a given line can be found using the formula shown, where $A x+B y+C=0$ is the equation of the line in standard form $(A, B, \text { and } C$ are integers).
Find the value(s) for $y$ that ensure $(1, y)$ is this same distance from $y=-\frac{1}{2} x+3$

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Key Concepts

Perpendicular Distance Formula

This concept involves computing the shortest (orthogonal) distance from a point to a line using a well?defined formula. The distance is determined by taking the absolute value of the linear combination of the point’s coordinates with the line’s coefficients, and then dividing by the square root of the sum of the squares of these coefficients. This formula is widely used in analytic geometry to measure how far a point is from a given line.

Standard Form of a Line

In analytic geometry, representing a line in its standard form, Ax + By + C = 0, is crucial because it directly corresponds with the coefficients used in the perpendicular distance formula. Transforming a line expressed in other forms (like slope-intercept form) into standard form ensures consistency and simplifies the process of applying various analytical tools, such as finding distances or intersections.

Solving Absolute Value Equations

Absolute value equations arise naturally when dealing with distances since distances are always non-negative. When an equation includes an absolute value, the solution process involves considering both the positive and negative scenarios of the inner expression. This method is a fundamental algebraic technique that is used in a wide range of problems to ensure all valid solutions are found.

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