Two sides of a rhombus are along the lines, $x - y + 1 = 0$ and $7x - y - 5 = 0$. If its diagonals intersect at $(-1, -2)$, then which one of the following is a vertex of this rhombus?
- (-3 , -9)
- (-3 , -8)
- $\left( \frac{1}{3} , - \frac{8}{3} \right)$
- $\left( - \frac{10}{3} , - \frac{7}{3} \right)$
The Correct Option is C
Solution and Explanation
Coordinates of $A \equiv (1, 2)$
$\therefore$ Slope of $AE = 2$
$\Rightarrow $ Slope of $BD = - \frac{1}{2}$ $\Rightarrow $
E of $BD$ is $\frac{y + 2 }{ x +1 } = - \frac{1}{2}$
$\Rightarrow x + 2y + 5 = 0$
$\therefore$ Co-ordinates of $D = \left( \frac{1}{3} , \frac{-8}{3} \right)$
So, the correct option is (C): \(\left( \frac{1}{3} , - \frac{8}{3} \right)\)
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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