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.