Question

If the circles touch both axes and pass through $\left( {2,3} \right)$ then find the length of the common chord.
A. $\sqrt 2 $
B. $2\sqrt 2 $
C. $3\sqrt 2 $
D. $4\sqrt 2 $

Answer 1

Hint: It is said in the question that two circles pass through $\left( {2,3} \right)$ and touch both axes. Since the two circles touch both the axes, so the coordinates of the centers of the two circles will be of the form $\left( {h,h} \right)$ and $\left( {k,k} \right)$. Now find out the equation of the line joining the centers of the two circles using two point form and the equation of the common chord joining point slope form. Then find the point of intersection of the two lines and find the equation of any of the two circles and solve the equation of the circle and the common chord to find the other endpoint of the common chord. After that using the distance formula, you’ll get the required length.

Formula Used:
Equation of a line passes through the points $\left( {{x_1},{y_1}} \right)$ and $\left( {{x_2},{y_2}} \right)$ is $\dfrac{{y - {y_1}}}{{x - {x_1}}} = \dfrac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}}$
The distance between two points $\left( {{x_1},{y_1}} \right)$ and $\left( {{x_2},{y_2}} \right)$ is $\sqrt {{{\left( {{x_2} - {x_1}} \right)}^2} + {{\left( {{y_2} - {y_1}} \right)}^2}} $ units.
If ${m_1}$ and ${m_2}$ be the slopes of the two perpendicular lines then ${m_1}{m_2} = - 1$
Equation of a line having slope $m$ and passing through the point $\left( {{x_1},{y_1}} \right)$ is $\dfrac{{y - {y_1}}}{{x - {x_1}}} = m$

Complete step by step solution:
It is said that the two circles touch both axes and pass through $\left( {2,3} \right)$.
Since, these circles touch both axes, so the distance of the centers of both the circles from the axes are equal.
Let these distances are $h$ for the smaller circle and $k$ for the bigger circle.
So, coordinates of the two circles may be taken in the form $\left( {h,h} \right)$ and $\left( {k,k} \right)$ respectively and radii are $h$ units and $k$ units respectively.
Now, find the equation of the line joining the centers $\left( {h,h} \right)$ and $\left( {k,k} \right)$.
Equation of the line is $\dfrac{{y - h}}{{x - h}} = \dfrac{{k - h}}{{k - h}}$
Simplify this equation.
Cancel the term $\left( {k - h} \right)$ from the numerator and the denominator of the right-hand side of the equation.
Because $h \ne k$. If $h = k$ then the two centers coincide and hence no such two circles exist.
$ \Rightarrow \dfrac{{y - h}}{{x - h}} = 1$
$ \Rightarrow y - h = x - h$
Cancel the term $h$ from both sides.
$ \Rightarrow y = x - - - - - \left( i \right)$
This is the equation of the line joining the two centers of the circles.
Now, the common chord is perpendicular to the line $\left( i \right)$.
Slope of the line $\left( i \right)$ is $1$.
So, slope of the common chord is $\left( { - 1} \right)$ and it passes through the point $\left( {2,3} \right)$
Hence, equation of the common chord is
$\dfrac{{y - 3}}{{x - 2}} = - 1$
Simplify this equation.
$ \Rightarrow y - 3 = - x + 2\\ \Rightarrow y = - x + 2 + 3\\ \Rightarrow y = 5 - x - - - - - \left( {ii} \right)$
Now, find the equation of any of the circles.
Let us find the point of intersection of the line $\left( i \right)$ and the common chord $\left( {ii} \right)$.
Solve these two equations to find the coordinates of the point.
Substituting $y = x$ in equation $\left( {ii} \right)$ is
$x = 5 - x$
Solve this equation to find the value of $x$
$ \Rightarrow x + x = 5\\ \Rightarrow 2x = 5\\ \Rightarrow x = \dfrac{5}{2}$
Put the value of $x$ in equation $\left( i \right)$ to find the value of $y$.
We get $y = \dfrac{5}{2}$
So, the point of intersection of the line $\left( i \right)$ and the line $\left( {ii} \right)$ is $\left( {\dfrac{5}{2},\dfrac{5}{2}} \right)$
This point bisects the common chord.
One end point of the chord is given as $\left( {2,3} \right)$
Let the other end point be $\left( {a,b} \right)$
Then the midpoint of the common chord is $\left( {\dfrac{{2 + a}}{2},\dfrac{{3 + b}}{2}} \right)$
So, $\dfrac{{2 + a}}{2} = \dfrac{5}{2}$ and $\dfrac{{3 + b}}{2} = \dfrac{5}{2}$
Solve these two equations to find the value of $a$ and $b$.
Solving first equation, we get $2 + a = 5 \Rightarrow a = 5 - 2 = 3$
Solving second equation, we get $3 + b = 5 \Rightarrow b = 5 - 3 = 2$
So, the other endpoint of the common chord is $\left( {3,2} \right)$
Now, find the length of the common chord i.e. the distance between the points $\left( {2,3} \right)$ and $\left( {3,2} \right)$
Using the distance formula, we get
The distance $ = \sqrt {{{\left( {3 - 2} \right)}^2} + {{\left( {2 - 3} \right)}^2}} = \sqrt {{{\left( 1 \right)}^2} + {{\left( { - 1} \right)}^2}} = \sqrt {1 + 1} = \sqrt 2 $ units.

Option ‘A’ is correct

Note: You can also find the coordinates of the other end point of the common chord directly by using the concept of reflection. First find the equation of the line passing through the centers of the two circles, which is $y = x$. The other endpoint is the reflection or image of the given endpoint $\left( {2,3} \right)$. So, the coordinates of the other endpoint will be $\left( {3,2} \right)$ and hence find the distance between these two endpoints which is equal to the length of the common chord.