Question

Let $A(x_1,y_1)$ and $B(x_2,y_2)$ are two fixed points.Then locus of point $P$ such that

• A. $\angle APB={90}^{\text{o}}$ is a circle
• B. Area of $△APB$ is minimum is a parabola
• C. $AP+PB=K$ where $K<AB$ is an ellipse
• D. $\frac{AP}{PB}=K$ where $K<1$ is a circle

Answer 1

Answer: (A), (D)

The correct options are
A $\angle APB={90}^{\text{o}}$ is a circle
D $\frac{AP}{PB}=K$ where $K<1$ is a circle

$\left(1\right)\angle APB={90}^{\text{o}}\Rightarrow \left(\text{slope of}AP\right)\cdot \left(\text{slope of}PB\right)=-1$

$\Rightarrow \left(\frac{y-y_1}{x-x_1}\right)\left(\frac{y-y_2}{x-x_2}\right)=-1$
$\Rightarrow (x-x_1)(x-x_2)+(y-y_1)(y-y_2)=0$
which is a circle

$\left(2\right)$ Area of $△APB$ is minimum
$\Rightarrow △APB=0$
$\Rightarrow A,P,B$ are collinear

$\left(3\right)AP+PB=K;\text{if}k=AB:\text{Straight line}$
$k>AB:\text{an ellipse}$
$k<AB:\text{no solution}$

$\left(4\right)\frac{AP}{PB}=K\text{If}k=1$

then locus is straight line and if $k<1$ or $k>1$ then locus is circle