The joint equation of lines passing through the origin and trisecting the first quadrant is ________
- $x^{2}+\sqrt{3}xy -y^{2} = 0$
- $x^{2}-\sqrt{3}xy -y^{2} = 0$
- $\sqrt{3}x^{2} -4xy +\sqrt{3}y^{2} = 0$
- $3x^{2} -y^{2} = 0 $
The Correct Option is C
Solution and Explanation
$\therefore$ Equation of line $OB$ is
$y=tan \,30^{\circ} x$
$ \Rightarrow y =\frac{1}{\sqrt{3}} x$
$x-\sqrt{3} y =0$
And equation of line $OC$ is
$y=tan \,60^{\circ} x$
$ \Rightarrow y =\sqrt{3x} $
$(\sqrt{3} x-y) =0$
$\therefore $ Combined equation is
$(x-\sqrt{3} y)(\sqrt{3} x-y)=0$
$\Rightarrow \sqrt{3} x^{2}-xy-3xy+\sqrt{3} y^{2}=0$
$\Rightarrow \sqrt{3} x^{2}-4xy+\sqrt{3} y^{2}=0$
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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