Question

If the point P(p, q) is equidistant from the points A(a+b, b-a) and B(a-b, a+b), then $\frac{p}{q}$ = ___

• A. $\frac{b}{a}$
• B. $\frac{-b}{a}$
• C. $\frac{-a}{b}$
• D. $\frac{a}{b}$

Answer 1

The correct option is D

$\frac{a}{b}$

Given that AP = BP
$\therefore A{P}^2=B{P}^2$

$\because$ Distance between points $(x_1,y_1)$ and$(x_2,y_2)$ is $\sqrt{(x_2-x_1{)}^2+(y_2-y_1{)}^2}$.
$\Rightarrow (a+b-p{)}^2+(b-a-q{)}^2=(a-b-p{)}^2+(a+b-q{)}^2\Rightarrow a^2+b^2+p^2+2ab-2bp-2ap+b^2+a^2+q^2-2ab+2aq-2bq=a^2+b^2+p^2-2ab+2bp-2ap+a^2+b^2+q^2+2ab-2bq-2aq$

$\Rightarrow 2ab-2bp-2ap-2ab+2aq-2bq=-2ab+2bp-2ap+2ab-2bq-2aq$

$\Rightarrow -4bp+4qa=0\Rightarrow -bp+aq=0$

$\Rightarrow bp=aq$

$\Rightarrow \frac{p}{q}=\frac{a}{b}$