Question

How do you write the equation of a circle given centre \[\left( {3, - 7} \right)\] and tangent to the \[y\] axis \[?\]

Answer 1

Hint: We need to know the basic equation for a circle and we need to know which is radius and which is central in the basic equation of a circle. We need to draw a diagram with the help of given information and basic formula for a circle. This question involves the operation of addition/ subtraction/ multiplication/ division. Also, we need to know the definition of the tangent line.

Complete step-by-step answer:
In the given question we have the following details,
The centre of a circle is \[\left( {3, - 7} \right)\] and the tangent line of the circle is \[y\] the axis.
The basic equation for a circle is,
 \[{\left( {x - h} \right)^2} + {\left( {y - k} \right)^2} = {r^2} \to \left( 1 \right)\]
Here, \[r\] is the value of radius and \[\left( {h,k} \right)\] is the centre of a circle.
So, we know that,
In the given question they give the centre as \[\left( {3, - 7} \right)\] from the formula we have the centre as \[\left( {h,k} \right)\] . So, we find the value of \[h\] and \[k\] as,
 \[h = 3\] and \[k = - 7\] .
So, let’s substitute the value of \[h\] and \[k\] in the equation \[\left( 1 \right)\] , we get
 \[\left( 1 \right) \to {\left( {x - h} \right)^2} + {\left( {y - k} \right)^2} = {r^2}\]
 \[{\left( {x - 3} \right)^2} + {\left( {y + 7} \right)^2} = {r^2} \to \left( 2 \right)\]
Next, we would know the definition of a tangent. Tangent is a line which touches the circle sharply at one point. In the given question, we have tangent to the \[y\] axis. From the value of the centre and tangent, we can draw the following diagram,
In the figure, we have a centre at the point \[\left( {3, - 7} \right)\] . We know that the \[y\] axis touches a circle at one point. by using these points we can draw the above figure. From the figure, we get the radius of the circle is \[3\] . So, the equation \[\left( 2 \right)\] becomes,
 \[\left( 2 \right) \to {\left( {x - 3} \right)^2} + {\left( {y + 7} \right)^2} = {r^2}\]
 \[{\left( {x - 3} \right)^2} + {\left( {y + 7} \right)^2} = {3^2}\]
So, the final answer is,
The equation of a given circle is,
 \[{\left( {x - 3} \right)^2} + {\left( {y + 7} \right)^2} = 9\] .
So, the correct answer is “ \[{\left( {x - 3} \right)^2} + {\left( {y + 7} \right)^2} = 9\] ”.

Note: This question involves the operation of addition/ subtraction/ multiplication/ division. We would remember the basic equation of a circle. Note that \[\left( {h,k} \right)\] is the centre of a circle and \[r\] is the radius of the circle. Note that the tangent line must touch at least one point on the circle. We would remember the square value of basic terms.