Question

A circle is centered at c(-5,4)and has a radius of 2. Find the equation of a concentric circle with half the radius.

Answer 1

Answer:  The required equation of the concentric circle is

$x^2+y^2+10x-8y+40=0.$

Step-by-step explanation:  Given that a circle is centered at the point C(-5,4) and has a radius of 2 units.

We are to find the equation of a concentric circle with half the radius.

We know that

the standard equation of a circle with center at the point (h, k) and radius 'r' units is given by

$(x-h)^2+(y-k)^2=r^2~~~~~~~~~~~~~~~~~~~~~~~(i)$

Since the centers of two concentric circles are same, so the center of the circle will be

center, (h, k) = (-5, 4)  and  radius is equal to half of 2 units.

So, radius of the concentric circle will be

$r=\dfrac{2}{2}=1~\textup{unit}.$

Therefore, the equation of the concentric circle is

$(x-h)^2+(y-k)^2=r^2\\\\\Rightarrow (x-(-5))^2+(y-4)^2=1^2\\\\\Rightarrow (x+5)^2+(y-4)^2=1\\\\\Rightarrow x^2+10x+25+y^2-8y+16=1\\\\\Rightarrow x^2+y^2+10x-8y+40=0.$

Thus, the required equation of the concentric circle is

$x^2+y^2+10x-8y+40=0.$

Answer 2

The answer to this is (x+5)^2+(y-4)^2=1