Question

The distance of a point $(x_1,y_1)$ from each of two straight lines which passes through the origin of co-ordinates is $\delta$; find the combined equation of these straight lines

• A. $(y_1^2-{\delta }^2)x^2-2x_1{y}_{1}xy+(x_1^2-{\delta }^2)y^2=0$
• B. $(y_1^2-{\delta }^2)x^2-x_1{y}_{1}xy+(x_1^2-{\delta }^2)y^2=0$
• C. $2(y_1^2-{\delta }^2)x^2-2x_1{y}_{1}xy+(x_1^2-{\delta }^2)y^2=0$
• D. $(y_1^2-{\delta }^2)x^2-2x_1{y}_{1}xy+2(x_1^2-{\delta }^2)y^2=0$

Answer 1

The correct option is A $(y_1^2-{\delta }^2)x^2-2x_1{y}_{1}xy+(x_1^2-{\delta }^2)y^2=0$

Let the general equation of the straight lines be $y=ax$ where $a$ is the slope and the parameter for the lines.

Distance of $(x_1,y_1)$ from $y=ax$ will be

$=\frac{|y_1-a{x}_1|}{\sqrt{1+a^2}}=\delta$

Or

$(y_1-a{x}_1{)}^2={\delta }^2(1+a^2)$

$y_1^2+a^2{x}_1^2-2a{y}_1{x}_1={\delta }^2+a^2{\delta }^2$

$(y_1^2-{\delta }^2)+a^2(x_1^2-{\delta }^2)-2a{x}_1{y}_1=0$

Now

$a=\frac{y}{x}$

Substituting in the above equation, we get

$x^2(y_1^2-{\delta }^2)+y^2(x_1^2-{\delta }^2)-2x{x}_{1}y{y}_1=0$