Question

A circle $C$ whose radius is $3$ ,touches externally the circle , ${x^2} + {y^2} + 2x - 4y - 4 = 0$ at the point $\left( {2,2} \right)$ , then the length of the intercept cut by this circle $C$, on the $x$ axis is equal to:
A) $2\sqrt 3$
B) $3\sqrt 2$
C) $\sqrt 5$
D) $2\sqrt 5$

Answer 1

Use the general formula for the circle to find the radius of the other circle. Observe that the two circles touch externally that means the sum of the radii is the distance between the centres of the circle. Use the formula to find the $x$ intercept for the circle $C$.

Complete step-by-step answer:
The data given in the problem is,
The radius of the circle $C$ is $3$.
${x^2} + {y^2} + 2x - 4y - 4 = 0$ at the point $\left( {2,2} \right)$ .
The general equation of a circle is ${x^2} + {y^2} + 2gx + 2hy + c = 0$ .
The radius of the above circle is given by $r = \sqrt {{g^2} + {f^2} - c}$.
The centre of the circle is given by $( - g, - h)$.
Comparing the given equation with the general equation of the circle we observe that $g = 1,h = - 2,c = - 4$.
Using the formula for the radius of the circle, we calculate the radius of the circle as follows:
$r = \sqrt {{1^2} + {{( - 2)}^2} - ( - 4)}$
Therefore, $r = 3$ .
Therefore, we observe that the radii of two circles are the same.
The centre of the other circle is $\left( { - 1, - \left( { - 2} \right)} \right) = \left( { - 1,2} \right)$.
It is given that both circles touch each other externally at the point $\left( {2,2} \right)$.
Therefore, the point $\left( {2,2} \right)$ is the midpoint of the line segment joining both the centres.
Also, the length of the segment joining both centres is $3 + 3 = 6$.
Let the centre of the circle $C$ be $\left( {a,b} \right)$.
We will use the midpoint formula to find the coordinates of the centre of the circle $C$.
$\left( {\dfrac{{ - 1 + a}}{2},\dfrac{{2 + b}}{2}} \right) = \left( {2,2} \right)$
Comparing both sides, we get $\left( {a,b} \right) = \left( {5,2} \right)$ .
Thus, the centre of the circle $C$ is $\left( {5,2} \right)$.
Equation of the circle $C$ is given as follows:
${\left( {x - 5} \right)^2} + {\left( {y - 2} \right)^2} = {3^2}$
Expanding the bracket and simplifying further we get ${x^2} + {y^2} - 10x - 4y + 20 = 0$.
Comparing with general equation of the circle we get $g = - 5,h = - 2,c = 20$ .
The $x$ intercept is given by $2\sqrt {{g^2} - c}$ .
Therefore, $x$ intercept for circle $C$ is given as follows:
$2\sqrt {{g^2} - c} = 2\sqrt {{{\left( { - 5} \right)}^2} - 20}$
Therefore, on simplifying, we obtain that the intercept at the $x$ axis is $2\sqrt 5$.

So, the correct answer is “Option D”.

Note: We used the general equation of the circle and not the standard equation of the circle. The circles touch each other externally at a single point therefore, the sum of the radii is the distance between the centres of the circle. We calculate the $x$ intercept for circle $C$ by writing the equation of circle $C$ in general form.