Question:

What is the image of the point (2, 3) in the line y = - x ?

Updated On: May 18, 2023

(- 3, - 2)
(- 3, 2)
(- 2, - 3)
(3, 2)

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The Correct Option is A

Solution and Explanation

Let there be a point P(2,3) on cartesian plane. Image of this point in the line y = -x will lie on a line which is perpendicular to this line and distance of this point from y = -x will be equal to distance of the image from this line. Let Q be the image of p and let the co-ordinate of Q be (b, k) Slope of line y = -x is -1 Line joining P, Q will be perpendicular to y = -x so, its slope = 1. Let the equation of the line be y = x + c since this passes through point (2, 3) 3 = 2 + c $\rightarrow$ c = 1 and the equation y = x + 1 The point of intersection R lies in the middle of P & Point of intersection of line y = - x and y = x + 1 is $2 y = 1 , \Rightarrow \, y = \frac{1}{2}$ and $x = - \frac{1}{2}$ Hence, $\frac{h +2}{2} = - \frac{1}{2}$ and $\frac{k + 3}{2} = \frac{1}{2}$ $\Rightarrow \, h = -3$ and $k = - 2$ So, the image of the point (2, 3) in the y = - x is ( -3, -2).

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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