Question

Which of the following is an equation of a circle in the $xy$-plane with center $(0,4)$ and a radius with endpoint $(\frac{4}{3},5)$?

• A. $x^2+{(y-4)}^2=\frac{25}{9}$
• B. $x^2+{(y+4)}^2=\frac{25}{9}$
• C. $x^2+{(y-4)}^2=\frac{5}{3}$
• D. $x^2+{(y+4)}^2=\frac{3}{5}$

Answer 1

Answer: (A)

$x^2+{(y-4)}^2=\frac{25}{9}$

Given, center of circle is $(0,4)$ and point on circle $(\frac{4}{3},5)$

$\therefore$ radius of circle $=\sqrt{{(\frac{4}{3}-0)}^2+(5-4{)}^2}$

$=\sqrt{\frac{16}{9}+1}=\sqrt{\frac{25}{9}}$

Since, Equation of the circle having centre $(x_1,y_1)$ and radius '$r$' is

$(x-x_1{)}^2+(y-y_1{)}^2=r^2$

$\therefore$ Equation of given circle $=(x-0{)}^2+(y-4{)}^2={\left(\sqrt{\frac{25}{9}}\right)}^2$

$=x^2+(y-4{)}^2=\frac{25}{9}$

Hence, option A is correct.