Question

The point(2,4) is joined to any point of the circle x²+y²=4. The locus of midpoint of AP as P moves on circle is​

Answer 1

Answer:

$\textbf{The locus of midpoint of AP as P moves on the circle is}\bf\,x^2+y^2-2x-4y+4=0$

Step-by-step explanation:

$\text{Let P(a,b) be any point on the circle }x^2+y^2=4$

$\text{Let (h,k) be the midpoint of the line joining A and P}$

$\text{Then,}$

$(h,k)=(\frac{2+a}{2},\frac{4+b}{2})$

$\implies\,h=\frac{2+a}{2}\:\text{ and }\:k=\frac{4+b}{2}$

$\implies\,2h=2+a\:\text{ and }\:2k=4+b$

$\implies\,a=2h-2\:\text{ and }\:b=2k-4$

$\text{since (a,b) lies on the circle, we have }a^2+b^2=4$

$\implies\,(2h-2)^2+(2k-4)^2=4$

$\implies\,4(h-1)^2+4(k-2)^2=4$

$\implies\,(h-1)^2+(k-2)^2=1$

$\implies\,h^2+1-2h+k^2+4-4k=1$

$\implies\,h^2+k^2-2h-4k+4=0$

$\therefore\textbf{The locus of midpoint of AP as P moves on the circle is }$

$\boxed{\bf\,x^2+y^2-2x-4y+4=0}$