Find the new coordinates of the point $(1, 1)$ if the origin is shifted to the point $(-3, -2)$ by a translation of axes.

The coordinates of new origin are $h = - 3$ , $k =-2$ and original coordinates are given by $x = 1$ , $y = 1$ . The transformation relation between the old coordinates $(x, y)$ and the new coordinates $(x', y')$ are given by $x = x' + h$ $\Rightarrow x'=x-h$ $y = y' +k$ $\Rightarrow y' =y-k$ Substituting the values, we have $x'= 1 + 3 = 4$ , $y' = 1 + 2 =3$ Hence, the coordinates of the point $(1, 1)$ in the new system are $(4,3)$ .

Find the new coordinates of the point $(1, 1)$ if

Notes on Straight lines

Concepts Used:

Straight lines

A straight line is a line having the shortest distance between two points.

A straight line can be represented as an equation in various forms, as show in the image below:

The following are the many forms of the equation of the line that are presented in straight line-

1. Slope – Point Form

Assume P0(x0, y0) is a fixed point on a non-vertical line L with m as its slope. If P (x, y) is an arbitrary point on L, then the point (x, y) lies on the line with slope m through the fixed point (x0, y0) if and only if its coordinates fulfil the equation below.

y – y₀ = m (x – x₀)

2. Two – Point Form

Let's look at the line. L crosses between two places. P₁(x₁, y₁) and P₂(x₂, y₂) are general points on L, while P (x, y) is a general point on L. As a result, the three points P₁, P₂, and P are collinear, and it becomes

The slope of P₂P = The slope of P₁P₂, i.e.

$\frac{y-y_1}{x-x_1} = \frac{y_2-y_1}{x_2-x_1}$

Hence, the equation becomes:

y - y₁ =$ \frac{y_2-y_1}{x_2-x_1} (x-x1)$

3. Slope-Intercept Form

Assume that a line L with slope m intersects the y-axis at a distance c from the origin, and that the distance c is referred to as the line L's y-intercept. As a result, the coordinates of the spot on the y-axis where the line intersects are (0, c). As a result, the slope of the line L is m, and it passes through a fixed point (0, c). The equation of the line L thus obtained from the slope – point form is given by

y – c =m( x - 0 )

As a result, the point (x, y) on the line with slope m and y-intercept c lies on the line, if and only if

y = m x +c