Question

Find the equation of a circle which passes through origin and cuts off intercepts $ - 2$ and $3$ from the axes.

Answer 1

Hint: Given, circle which passes through origin and cuts off intercepts $ - 2$ and $3$. We have to find the equation of the circle. First, we will put the given points in the general equation of the circle to find the unknown variable. After calculating those unknown we will put them in the general equation to get the required equation of the circle.

Formula Used: General equation of circle:
${x^2} + {y^2} + 2gx + 2fy + c = 0$

Complete step by step solution: A circle is a closed curve that extends outward from a set point known as the centre, with each point on the curve being equally spaced from the centre. ${\left( {x - h} \right)^2} + {\left( {y - k} \right)^2} = {r^2}$ is the equation for a circle having a centre (h, k) and $r$ radius.
This is the equation's standard form. Thus, we can quickly determine the equation of a circle if we know its radius and centre coordinates.
Given, circle which passes through origin and cuts off intercepts $ - 2$ and $3$.
We know the general equation of circle.
${x^2} + {y^2} + 2gx + 2fy + c = 0$ (1)
Clearly, given the circle passes through points
$C\left( {0,0} \right),\,P( - 2,0),\,Q(0,3)$
Putting the point C that is $x = 0$ and $y = 0$ in the equation (1)
$0 + 0 + 0 + 0 + c = 0$
$ \Rightarrow c = 0$
Putting the point P that is $x = - 2$ and $y = 0$ in the equation (1)
${( - 2)^2} + 0 + 2( - 2)g + 0 = 0$
$4 - 4g = 0$
Shifting 4 to other side.
$4g = 4$
Dividing both sides with 4
$g = 1$
Putting point Q that is $x = 0$ and $y = 3$ in the equation (1)
$0 + {(3)^2} + 0 + 2(3)f = 0$
$9 + 6f = 0$
Shifting 9 to other side
$6f = - 9$
Dividing both sides with 6
$f = - \dfrac{3}{2}$
Putting value of g, f and c in the equation (1)
${x^2} + {y^2} + 2(1)x + 2\left( {\dfrac{{ - 3}}{2}} \right)y + 0 = 0$
${x^2} + {y^2} + 2x - 3x = 0$
Hence, ${x^2} + {y^2} + 2x - 3x = 0$ is the required equation of the circle.

Note: Students should solve questions carefully to avoid any calculation error or any conceptual error. They should pay attention to the information which is given in the question. Ans should use the general equation of the circle that is ${x^2} + {y^2} + 2gx + 2fy + c = 0$ to get the correct answer without any difficulty.