Question

The equation of the lines passing through the point (1, 0) and at a distance from the origin are None of these

Answer 1

x - √3y - 1 = 0 and x + √3y - 1 = 0 (assuming distance is 1/2)

Answer 2

The problem asks for the equation of lines passing through the point (1, 0) and at a certain distance from the origin. The distance value is missing in the question statement. Assuming a common value for such problems, let's consider the distance to be $\frac{1}{2}$.

Let the equation of a line passing through the point $(1, 0)$ be $y - y_1 = m(x - x_1)$. Substituting $(x_1, y_1) = (1, 0)$: $y - 0 = m(x - 1)$ $y = mx - m$ Rearranging into the standard form $Ax + By + C = 0$: $mx - y - m = 0$

The distance $d$ of a line $Ax + By + C = 0$ from the origin $(0, 0)$ is given by the formula: $d = \frac{|A(0) + B(0) + C|}{\sqrt{A^2 + B^2}} = \frac{|C|}{\sqrt{A^2 + B^2}}$

In our case, $A = m$, $B = -1$, $C = -m$. Given that the distance $d = \frac{1}{2}$ (assuming this missing value): $\frac{1}{2} = \frac{|-m|}{\sqrt{m^2 + (-1)^2}}$ $\frac{1}{2} = \frac{|m|}{\sqrt{m^2 + 1}}$

To solve for $m$, square both sides of the equation: $\left(\frac{1}{2}\right)^2 = \left(\frac{|m|}{\sqrt{m^2 + 1}}\right)^2$ $\frac{1}{4} = \frac{m^2}{m^2 + 1}$

Cross-multiply: $1 \cdot (m^2 + 1) = 4 \cdot m^2$ $m^2 + 1 = 4m^2$ $1 = 4m^2 - m^2$ $1 = 3m^2$ $m^2 = \frac{1}{3}$ $m = \pm\sqrt{\frac{1}{3}}$ $m = \pm\frac{1}{\sqrt{3}}$

Now substitute these two values of $m$ back into the line equation $y = m(x - 1)$:

Case 1: $m = \frac{1}{\sqrt{3}}$ $y = \frac{1}{\sqrt{3}}(x - 1)$ Multiply by $\sqrt{3}$: $\sqrt{3}y = x - 1$ Rearrange: $x - \sqrt{3}y - 1 = 0$

Case 2: $m = -\frac{1}{\sqrt{3}}$ $y = -\frac{1}{\sqrt{3}}(x - 1)$ Multiply by $\sqrt{3}$: $\sqrt{3}y = -(x - 1)$ $\sqrt{3}y = -x + 1$ Rearrange: $x + \sqrt{3}y - 1 = 0$

Thus, the two equations of the lines are $x - \sqrt{3}y - 1 = 0$ and $x + \sqrt{3}y - 1 = 0$.

The question statement includes "None of these" as if it were an option. If the derived equations are not listed among typical options, then "None of these" would be the correct choice. Since no other options are provided, we state the derived equations.