Question

For the line 3x + 2y = 12 and the circle \[\begin{array}{*{20}{c}}
  {{x^2} + {y^2} - 4x - 6y + 3}& = &0
\end{array}\], which of the following statements is true?
A) Line is a tangent to a circle
B) Line is a chord of a circle
C) Line is a diameter of a circle
D) None of these

Answer 1

Hint: First of all, we will determine the coordinates of the circle. After getting the coordinates of the circle, we will satisfy these coordinates with the given line. If the coordinates satisfy the line, then we will get a desired answer.

Complete Step by step solution:
In this question, we have given the equation of the circle and the line. With the help of this information, we will have to find the suitable answer. For that purpose, we will find the coordinates of the circle. Therefore,
\[\begin{array}{*{20}{c}}
  { \Rightarrow {x^2} + {y^2} - 4x - 6y + 3}& = &0
\end{array}\]
Now convert this equation into the general equation of the circle, for that purpose we will add and subtract 10 in the above equation. Therefore, we will get
\[\begin{array}{*{20}{c}}
  { \Rightarrow {x^2} + {y^2} - 4x - 6y + 3 + 10 - 10}& = &0
\end{array}\]
\[\begin{array}{*{20}{c}}
  { \Rightarrow \left( {{x^2} + 4 - 4x} \right) + \left( {{y^2} + 9 - 6y} \right)}& = &{10}
\end{array}\]
\[\begin{array}{*{20}{c}}
  { \Rightarrow {{\left( {x - 2} \right)}^2} + {{\left( {y - 3} \right)}^2}}& = &{10}
\end{array}\]
On comparing the above equation with \[\begin{array}{*{20}{c}}
  {{{\left( {x - a} \right)}^2} + {{\left( {y - b} \right)}^2}}& = &{{r^2}}
\end{array}\],we will get
\[\begin{array}{*{20}{c}}
  { \Rightarrow a}& = &2
\end{array}\] and \[\begin{array}{*{20}{c}}
  b& = &3
\end{array}\]
Therefore, the center of the coordinate is (2, 3)
Now satisfy these coordinates with the equation 3x +2y = 12.
\[\begin{array}{*{20}{c}}
  { \Rightarrow 3a + 2b}& = &{12}
\end{array}\]
\[\begin{array}{*{20}{c}}
  { \Rightarrow 3 \times 2 + 2 \times 3}& = &{12}
\end{array}\]
\[\begin{array}{*{20}{c}}
  { \Rightarrow 12}& = &{12}
\end{array}\]
Now these coordinates satisfy the equation. It means the circle touches the line.
Therefore, the line will be tangent to a circle.
Therefore, the correct option is (A)

Note: It is important to note that if the coordinates of the circle satisfy the line, then the line is said to be tangent to the circle. If any line passes through the circle, then it means that the coordinates of the line must satisfy the equation of the circle or there must be a relation between the line and the circle.