Question

If a point $P$ moves such that the distance from the point $A(1,1)$ and the line $x+y+2=0$ are equal then the locus of $P$ is equal to

• A. a straightline
• B. a parabola
• C. a pair of st. lines
• D. an ellipse

Answer 1

Answer: (B)

a parabola

Let the coordinate at $P$ is $(h,k)$

$\therefore$ Distance com point $A(1,1)$ is $AP=\sqrt{(n-1{)}^2+(k-1{)}^2}$

$\therefore$ Distance com line $x+y+2=0$ is $=\frac{h+k+2}{\sqrt{1^2+1^2}}$

According to equation

$\sqrt{(h-1{)}^2+(k-1{)}^2}=\frac{h+k+2}{\sqrt{2}}$

Squaring both are

$2\left\{\right(h-1{)}^2+(k-1{)}^2\}=(h+k+2{)}^2$

$\Rightarrow 2(h^2-2h+1+k^2-2k+1)=h^2+k^2+4+2hk+4h+4k$

$\Rightarrow 2h^2-4h+2k^2-4k+4=h^2+k^2+4+2hk+4h+4k$

$\Rightarrow h^2-hk+k^2=8h+4k$

$\Rightarrow (h-k{)}^2=8(h+k)$

Locus of $P$ is, $(x-y{)}^2=8(x+y)$ which is a parabola

Option $\to \left(B\right)$