Question

A circle with centre at \[(2,\ 4)\] is such that the line \[x + y + 2 = 0\] cuts a chord of length \[6\]. The radius of the circle is
A \[\sqrt{41}\] cm
B \[\sqrt{11}\] cm
C \[\sqrt{21}\] cm
D \[\sqrt{31}\] cm

Answer 1

Hint: In this question, given a circle with centre at \[(2,\ 4)\] and the line \[x + y + 2 = 0\] cuts a chord, also the length of the chord is \[6\] . Before proceeding with this type of question, we must draw the circle with center \[(2,4)\] with the given above conditions so that we can understand the question more clearly. Then we need to compare the given line equation with \[Ax + By + C = 0\] to get the value of \[A\], \[B\] and \[C\]. Then we can use the formula of the length of the perpendicular from a point to the line. Further by using Pythagoras theorem, we can find the radius of the circle.

Complete step by step Solution:
Given a circle with centre at \[(2,\ 4)\] and the line \[x + y + 2 = 0\] cuts a chord, also the length of the chord is \[6\] .
Diagram :


First on comparing the equation of the line with \[Ax + By + C = 0\] ,
We get,
\[A = 1\] , \[B = 1\] and \[C = 2\]
The length of perpendicular from \[\left( x_{1},y_{1} \right)\] on the line \[Ax + By + C = 0\] will be \[\dfrac{Ax_{1} + By_{1} + C}{\sqrt{A + B}}\]
Now The length of perpendicular from \[(2,4)\] on the line \[x + y + 2 = 0\] will be \[\dfrac{\left( 1 \times 2 \right) + \left( 1 \times 4 \right) + 2}{\sqrt{1 + 1}}\]
On solving,
We get,
The length of perpendicular is \[\dfrac{2 + 4 + 2}{\sqrt{2}}\]
On further simplification,
We get, The length of perpendicular is \[\dfrac{8}{\sqrt{2}}\] which is \[4\sqrt{2}\]
Given the length of the chord is \[6\], the perpendicular from the center onto the chord bisects the chord. So, the base of the triangle will be \[3\].
Now the radius of the circle will be the hypotenuse of the right-angled triangle formed.
Now by using Pythagoras theorem,
We get,
The radius of the circle,
\[r = \sqrt{\left( 3 \right)^{2}\ + \left( 4\sqrt{2} \right)^{2}}\]
On simplifying,
We get,
\[r = \sqrt{9 + 16 \times 2}\]
On further simplification,
We get,
\[r = \sqrt{9 + 32}\]
On solving,
We get,
\[r = \sqrt{41}\] cm
Thus the radius of the circle is \[\sqrt{41}\] cm

Hence, the correct option is A.

Note: In order to solve these questions, we should have a strong grip over line equations, perpendicular distance, and Pythagoras theorem. We should keep in mind that we need to find the length of the perpendicular in order to form the triangle and find the radius of the circle. We can also directly find The length of perpendicular from \[\left( x_{1},y_{1} \right)\] on the line \[Ax + By + C = 0\] will be \[\dfrac{Ax_{1} + By_{1} + C}{\sqrt{A + B}}\] .