Question

If the distance of a point $(x_1,y_1)$ from each of two straight lines, which pass through the origin of coordinates, is $\delta$, then the two lines are given by

• A. ${(x_{1}y-x{y}_1)}^2={\delta }^2(x^2+y^2)$
• B. ${(x_{1}y+x{y}_1)}^2={\delta }^2(x^2+y^2)$
• C. ${(x_{1}y-x{y}_1)}^2={\delta }^2(x^2-y^2)$
• D. None of these

Answer 1

The correct option is A ${(x_{1}y-x{y}_1)}^2={\delta }^2(x^2+y^2)$

Any line through the origin is

$y=mx$ or $mx-y=0$.

$\because \delta =$ distance of $(x_1,y_1)$ from the line $mx-y=0$

$\therefore \delta =\pm \frac{{mx}_1-y_1}{\sqrt{1+m^2}}$ or ${\delta }^2(1+m^2)={({mx}_1-y_1)}^2$ ...(1)

But, $y=mx$ or $m=\frac{y}{x}$

$\therefore$ The required equation is

${\delta }^2(\frac{y^2}{x^2}+1)={(\frac{y}{x}.x_1-y_1)}^2$ $\Rightarrow {\delta }^2(x^2+y^2)={(x_{1}y-{xy}_1)}^2$