Question

A straight line is such that the algebraic sum of the perpendiculars drawn upon it from any number of fixed points is zero. Then the straight line ______.

• A. passes through a fixed point
• B. is parallel to a fixed line
• C. perpendicular to a fixed line
• D. None of these

Answer 1

Answer: (A)

passes through a fixed point

Let line

$ax+by+c=0$

Point $A(x_1,y_1),B(x_2,y_2)$

Then perpendicular distance from Point A

$d_1=\left|\frac{a{x}_1+b{y}_1+c}{\sqrt{a^2+b^2}}\right|$

and perpendicular distance from Point B

$d_2=\left|\frac{a{x}_2+b{y}_2+c}{\sqrt{a^2+b^2}}\right|$

according to question

$d_1+d_2=0$

$\frac{a{x}_1+b{y}_1+c}{\sqrt{a^2+b^2}}+\frac{a{x}_2+b{y}_2+c}{\sqrt{a^2+b^2}}=0$

$\frac{a{x}_1+b{y}_1+c+a{x}_2+b{y}_2+c}{\sqrt{a^2+b^2}}=0$

$a(x_1+x_2)+b(y_1+y_2)+2c=0$

This line passes through fixed point

$x=\frac{x_1+x_2}{2}$ and $y=\frac{y_1+y_2}{2}$