Question

If the line segments joining the points (a, b) and (c, d) subtends a right angle at the origin, then which of the equation is correct ?

• A. ac - bd = 0
• B. ac + bd = 0
• C. ab + cd = 0
• D. ab - cd = 0

Answer 1

Answer: (B)

ac + bd = 0

R.E.F image

Let the coordinates of 0 be $(0,0)$

p be $(a,b)$ & Q be $(c,d)$

from the figure above

$\angle POQ={90}^{\circ }$

i.e. $OP$ and $OQ$ lines are perpendicular

to each other

$\therefore$ The product of their slopes is $-1$

As we know, the slope of a line joining

two points be $(x_1,y_1)$ & $(x_2,y_2)$ is

given by7 $\frac{y_2-y_1}{x_2-x_1}$

So, putting the coordinates above, we

get

Slope of $OP=\frac{b-o}{a-o}=\frac{b}{a}$

& Slope of $OQ=\frac{d-o}{c-o}=\frac{d}{c}$

$\because$ The produced of the slopes is $-1$, So,

$\frac{b}{a}\times \frac{d}{c}=-1$

$bd=-ac\Rightarrow bd+ac=0$