Question

What is the standard form of the equation of a circle with centre at point $(5,8)$ and which passes through the point $(2,5)$?

Answer 1

In this question we have to find out the equation of the circle with the given two points one is the centre and another is the point on the circle. For finding the first we need to find out the radius of the circle which is nothing but the distance between the given two points. After finding the radius we just need to put that in the standard form of the equation of the circle.
Formula:
i)The shortest distance of two points $({x_1},{y_1})$ and $({x_2},{y_2})$is
$\sqrt {{{({x_2} - {x_1})}^2} + {{({y_2} - {y_1})}^2}}$
ii)The standard form of the equation of a circle is
${(x - h)^2} + {(y - k)^2} = {r^2}$
where,
$(h,k)$ is the centre of the circle and r is the radius of the circle.

Complete step-by-step solution:
We need to determine the standard form of the equation of a circle with centre at point $(5,8)$ and which passes through the point $(2,5)$.
First we need to find out the shortest distance of $(5,8)$ and $(2,5)$ which is nothing but the radius of the circle, since the straight line connecting the centre of the circle and the point on the circle is the radius of the circle.
The distance between the points $(5,8)$ and $(2,5)$

& = \sqrt {{{(2 - 5)}^2} + {{(5 - 8)}^2}} \\\ & = \sqrt {{{\left( { - 3} \right)}^2} + {{\left( { - 3} \right)}^2}} \\\ & = \sqrt {9 + 9} \\\ & = \sqrt {18} units \\\ \end{aligned} $$ Hence the radius of the circle r =$$\sqrt {18} $$ Now we know the standard form of the equation of a circle is $${(x - h)^2} + {(y - k)^2} = {r^2}$$ Where, $$(h,k)$$ is the centre of the circle and r is the radius of the circle. Using the above standard form for the given point, we get, The equation of the circle with centre at $$(5,8)$$ is $${(x - 5)^2} + {(y - 8)^2} = {\left( {\sqrt {18} } \right)^2}$$ Or,$${(x - 5)^2} + {(y - 8)^2} = 18$$ **Thus the standard form of the equation of a circle with centre is at point $$(5,8)$$ and which passes through the point $$(2,5)$$ is $${(x - 5)^2} + {(y - 8)^2} = 18$$.** **Note:** A circle is a shape consisting of all points in a plane that are a given distance from a given point, the centre; equivalently it is the curve traced out by a point that moves in a plane so that its distance from a given point is constant. Area of the circle is $$\pi {r^2}$$.