Question

Find the equation to the circle whose radius is \[\sqrt {{a^2} - {b^2}} \] and whose center is \[\left( { - a, - b} \right)\].

Answer 1

Hint: In the given question, we have been given that there is a circle whose center lies on the point \[\left( { - a, - b} \right)\] and whose radius follows the relation \[\sqrt {{a^2} - {b^2}} \]. We have to calculate the equation of the circle. For doing this, we are first going to write the equation of the circle in the standard form. Standard form means that the variables used in it are generic and can be replaced with the variables or constants given in the question. Then, we just need to replace the variables in the standard form with the ones given in the question.

Formula Used:
For this question, we are going to use the standard formula of the circle:
The equation of a circle with center at \[\left( {h,k} \right)\] and radius \[r\] is given by:
\[{\left( {x - h} \right)^2} + {\left( {y - k} \right)^2} = {r^2}\]

Complete step-by-step answer:
In this question, the radius is \[\sqrt {{a^2} - {b^2}} \] and the center is at \[\left( { - a, - b} \right)\].
For finding the equation of the given circle, we just need to put in these values in the standard form of the circle,
\[{\left( {x - h} \right)^2} + {\left( {y - k} \right)^2} = {r^2}\]
And we get,
$\Rightarrow$ \[{\left( {x - \left( { - a} \right)} \right)^2} + {\left( {y - \left( { - b} \right)} \right)^2} = {\left( {\sqrt {{a^2} - {b^2}} } \right)^2}\]
$\Rightarrow$ \[{\left( {x + a} \right)^2} + {\left( {y + b} \right)^2} = {a^2} - {b^2}\]
Opening the brackets and solving,
$\Rightarrow$ \[{x^2} + {a^2} + 2ax + {y^2} + {b^2} + 2by - {a^2} + {b^2} = 0\]
Cancelling out what are being cancelled,
$\Rightarrow$ \[{x^2} + 2ax + {y^2} + 2by + 2{b^2} = 0\]
Hence, the equation of the given circle is \[{x^2} + 2ax + {y^2} + 2by + 2{b^2} = 0\].

Note: So, for solving questions of such type, we first write what has been given to us. Then we write down what we have to find. Then we think about the formulae which contain the known and the unknown and pick the one which is the most suitable and the most effective for finding the answer of the given question. Then we put in the knowns into the formula, evaluate the answer and find the unknown. It is really important to follow all the steps of the formula to solve the given expression very carefully and in the correct order, because even a slightest error is going to make the whole expression awry and is going to give us an incorrect answer.