The equation of a line at a distance \(\sqrt{5}\) units from the origin and the ratio of the intercepts on the axes is \(1: 2\), is \(\ldots \ldots\) (a) \(2 \mathrm{x}+\mathrm{y} \pm 5=0\) (b) \(2 x+y \pm 5=0\) (c) \(x-2 y \pm 5=0\) (d) None of these

Coordinate geometry, also known as analytic geometry, is a branch of mathematics that brings together algebra and geometry through the use of a coordinate system. This system allows for the precise localization of points in two-dimensional space using a pair of numerical coordinates, usually represented as (x, y). These coordinates indicate the position of a point in relation to two fixed perpendicular lines, called axes, which are the x-axis (horizontal) and the y-axis (vertical).

With coordinate geometry, we can easily define lines, curves, and shapes using equations. For example, the line equation in intercept form, which is particularly useful in this context, can be expressed as \frac{x}{a} + \frac{y}{b} = 1\, where \(a\) and \(b\) are the x-intercept and y-intercept, respectively. This form is essential when we have information about a line's interaction with the coordinate axes, as it allows us to explore properties such as slope and distance from the origin, as well as the relative lengths of intercepts on the axes.

The distance from the origin to any point (x, y) on the Cartesian plane can be determined using the Euclidean distance formula, which is the square root of the sum of the squares of its coordinates, given by \( d = \sqrt{x^2 + y^2} \). When it comes to the distance from the origin to a line, however, the approach is slightly different.

For a straight line defined in intercept form \frac{x}{a} + \frac{y}{b} = 1\, we use the formula \( d = \frac{|c|}{\sqrt{A^2 + B^2}} \) where \( Ax + By + c = 0 \) is the general form of the line equation, and \(d\) is the shortest distance to the origin. Understanding this concept is crucial for solving our given problem, as we have the distance \(\sqrt{5}\) units, which will help us find the relationship between the intercepts when further applied to the intercept form equation.

When working with lines on a coordinate plane, the ratio of intercepts is an interesting property that gives us a sense of the line's orientation and its relative distance from the axes. If a line cuts the x-axis at \(a\) and the y-axis at \(b\), then the ratio \(\frac{a}{b}\) defines how the line divides the quadrants.

In our particular exercise, the ratio of intercepts is given as 1:2. To express this algebraically, we write the x-intercept as \(a = k\) and the y-intercept as \(b = 2k\). Combining this ratio with the information about the distance from the origin allows us to solve for the specific line equation that satisfies both conditions. By plugging these values of intercepts into the intercept form equation and utilizing the given distance from the origin, we derive the particular equation that corresponds to the given line characteristics.