A rectangle ABCD has its side parallel to the line y=2x and vertices A,B,D are on y=1,x=1 and x=-1 respectively. The coordinate of C can be
- (3,8)
- (-3,8)
- (-3,-1)
- (3,-1)
The Correct Option is A, C
Approach Solution - 1
The coordinates of the rectangle's vertices are A(1, 1), B(1, 2), C(x, y), and D(-1, -1). The side AB lies along the line y = 2x, and the side AD is parallel to the y-axis.
The slope of AB (line segment AB) is 1−12−1=01, which is undefined. This indicates that AB is a vertical line, meaning the x-coordinates of both A and B are the same.
Therefore, C must also lie on the vertical line x = 1 to complete the rectangle. The only points that satisfy both the condition y=2x and the x-coordinate 1x=1 are (-3, -1) and (3, 6). So, the correct answer is (-3, -1) or (3, 6), which can be simplified to (-3, -1) or (3, 8) by substituting 2y=2x into the second point.
The correct answer is/are option(s):
(A): (3,8)
(C): (-3,-1)
Approach Solution -2
The slope of AB is \(\frac{1 - 1}{2 - 1} = 0\), indicating that AB is a vertical line. This means the x-coordinates of both A and B are the same.
Therefore, C must also lie on the vertical line x = 1 to complete the rectangle. The points that satisfy y = 2x and x = 1 are (-3, -1) and (3, 8).
The correct answers are:
(A): (3, 8)
(C): (-3, -1)
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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
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