Question

If $A(2,0)$ and $B(0,2)$ are given points and $P$ is a point such that $PA:PB=2:3,$ then the locus of $P$ passes through the point $(a,a)$ for

• A. $a>2$
• B. $a=1$
• C. no real value of $a$.
• D. finite number of real values of $a$.

Answer 1

The correct option is C no real value of $a$.

Let $P(x,y)$ be such that $PA:PB=2:3$
then $9(PA{)}^2=4(PB{)}^2$
$\Rightarrow 9[{(x-2)}^2+y^2]=4[x^2+{(y-2)}^2]$
$\Rightarrow 9(x^2+y^2-4x+4)=4(x^2+y^2-4y+4)$
$\Rightarrow 5x^2+5y^2-36x+16y+20=0$
which is the locus of P and it passes through $(a,a)$ if
$5a^2+5a^2-36a+16a+20=0$
or $10a^2-20a+20=0$
or $(a-1{)}^2+1=0$
which is not possible for any real value of $a.$