Question

The equation of circle whose centre is \[(1, - 3)\] and which touches the line \[2{\rm{ }}x - y - 4 = 0\], is
A. \[5{x^2} + 5{y^2} + 10x + 30y + 49 = 0\]
B. \[5{x^2} + 5{y^2} + 10x - 30y - 49 = 0\]
C. \[5{x^2} + 5{y^2} - 10x + 30y - 49 = 0\]
D. None of these

Answer 1

Hint:To answer this question, we will assume that \[{x^2} + {y^2} - 2gx - 2fy + c = 0\] is the equation of the required circle. Following that, we will form equations between \[g,f,c\] and solve them to find their values using the given data and hence find the required equation of a circle with centre \[(1, - 3)\].

Complete step by step Solution:
We have been given that the circle passes through\[(1, - 3)\] and touches the equation
\[2{\rm{ }}x - y - 4 = 0\]
The general equation of a circle is
\[{(x - h)^2} + {(y - k)^2} = {r^2}\].
Here,\[(h,k)\]is the centre of the circle
 That is the centre of the circle is: \[(1, - 3)\].
It is given that circle with the centre \[(1, - 3)\] touches the straight line \[2x - y - 4 = 0\].
As a result, the length of the perpendicular dropped from centre \[\left( {1, - 3} \right)\] to the line.
Then \[2x - y - 4 = 0\] will be equal to the radius of the circle.
Radius \[ = \dfrac{{|2 \times 1 + 3 - 4|}}{{\sqrt {{2^2} + {{( - 1)}^2}} }}\]
First we have to simplify the numerator: \[ = \dfrac{1}{{\sqrt {{2^2} + {{( - 1)}^2}} }}\]
Now simplify the denominator\[{2^2} + {( - 1)^2} = 5\]: \[ = \dfrac{1}{{\sqrt 5 }}\]
Hence the equation of the circle is
\[{({\rm{x}} - 1)^2} + {({\rm{y}} - ( - 3))^2} = {\left( {\dfrac{1}{{\sqrt 5 }}} \right)^2}\]
Multiply the minus sign with the number inside the parentheses:
\[{(x - 1)^2} + {(y + 3)^2} = {\left( {\dfrac{1}{{\sqrt 5 }}} \right)^2}\]
Expand the above expression to make to less complicated:
\[ \Rightarrow {{\rm{x}}^2} - 2{\rm{x}} + 1 + {{\rm{y}}^2} + 6{\rm{y}} + 9 = \dfrac{1}{5}\]
Add the constants to make it simpler:
\[ \Rightarrow {{\rm{x}}^2} + {{\rm{y}}^2} - 2{\rm{x}} + 6{\rm{y}} + 10 = \dfrac{1}{5}\]
Multiply\[{\rm{5}}\]with equation on the left side:
\[ \Rightarrow 5{{\rm{x}}^2} + 5{{\rm{y}}^2} - 10{\rm{x}} + 30{\rm{y}} + 50 - 1 = 0\]
Simplify the like terms:
\[\therefore 5{{\rm{x}}^2} + 5{{\rm{y}}^2} - 10{\rm{x}} + 30{\rm{y}} + 49 = 0\]
Therefore,\[5{{\rm{x}}^2} + 5{{\rm{y}}^2} - 10{\rm{x}} + 30{\rm{y}} + 49 = 0\]is the required equation of the circle.

Therefore, the correct option is (D).


Note: The student should start with the basic equation of the circle. After that, try to correctly form the equations based on the data provided in the question. In order to get the correct answer, students should avoid making calculation errors while solving.