Question

Find the equation of the circle with centre at $(a,b)$ touching the Y-axis.

Answer 1

Hint: Here, we will find the equation of the circle by using the general formula of a circle. The equation of the circle is an expression that represents the circle when it touches any axis at a point. If it touches the vertical axis it is called an equation of circle touching $y$-axis and if it touches the horizontal axis it is called an equation of circle touching $x$-axis.

Formula used:

We will use the formula ${\left( {x - h} \right)^2} + {\left( {y - k} \right)^2} = {r^2}$ where $h$ is $x$ coordinate of circle point, $k$ is $y$ coordinate of center point and $r$ is radius of the circle. $\begin{array}{l}\\\\\end{array}$

Complete step-by-step answer:

We know that equation of circle is formed is given by

${\left( {x - h} \right)^2} + {\left( {y - k} \right)^2} = {r^2}$……………………………$\left( 1 \right)$

Now, value of the centre is given as $(a,b)$

Therefore we get the value of $h$ and $k$ as

$h = a$ and $k = b$

Next, we have that the circle is touching the y-axis.

So now we will draw the diagram of the circle based on the given information.

Therefore, the radius of the circle is $a$.

Now, we will substitute all the values in equation $\left( 1 \right)$. Therefore, we get

${\left( {x - a} \right)^2} + {\left( {y - b} \right)^2} = {a^2}$

So the equation of the circle is ${\left( {x - a} \right)^2} + {\left( {y - b} \right)^2} = {a^2}$.

Note: We need to keep in mind that if the circle touches $y$-axis the $x$ coordinate of center will be the radius of the circle similarly if circle touch $y$-axis the y coordinate of center will be the radius of the circle.

The mistake that can be made is that we can draw the diagram wrong by making the circle touch both the axis as it clearly mentioned that the circle touches the Y-axis. So we have to be very careful while finding the equation