Answer
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Hint: First, substitute the given points $\left( {0,0} \right),\left( {1,3} \right)$, and $\left( {2,4} \right)$ in the standard equation of a circle to calculate the values of $c,f$, and $g$. Then, simplify the standard equation of a circle by substituting the values. And in the end, substitute the point $\left( {k,3} \right)$ in the simplified equation of a circle and solve it to reach the required answer.
Formula Used:
The standard equation of a circle is: ${x^2} + {y^2} + 2gx + 2fy + c = 0$, where $\left( { - g, - f} \right)$ is the center of a circle.
Complete step by step solution:
Given: The points $\left( {0,0} \right),\left( {1,3} \right),\left( {2,4} \right)$, and $\left( {k,3} \right)$ are concyclic.
We know that if points are concyclic, then they lie on the same circle.
Substitute the point $\left( {0,0} \right)$ in the standard equation of a circle.
We get,
${0^2} + {0^2} + 2g\left( 0 \right) + 2f\left( 0 \right) + c = 0$
$ \Rightarrow c = 0$ $.....\left( 1 \right)$
Now substitute the point $\left( {1,3} \right)$ in the standard equation of a circle.
We get,
${1^2} + {3^2} + 2g\left( 1 \right) + 2f\left( 3 \right) = 0$
$ \Rightarrow 1 + 9 + 2g + 6f = 0$
$ \Rightarrow 10 + 2g + 6f = 0$ $.....\left( 2 \right)$
Now substitute the point $\left( {2,4} \right)$ in the standard equation of a circle.
We get,
${2^2} + {4^2} + 2g\left( 2 \right) + 2f\left( 4 \right) = 0$
$ \Rightarrow 4 + 16 + 4g + 8f = 0$
$ \Rightarrow 20 + 4g + 8f = 0$
Divide both sides by 2.
$10 + 2g + 4f = 0$
$ \Rightarrow 10 + 2g = - 4f$ $.....\left( 3 \right)$
Substitute the equation $\left( 3 \right)$ in the equation $\left( 2 \right)$.
$ - 4f + 6f = 0$
$ \Rightarrow 2f = 0$
$ \Rightarrow f = 0$ $.....\left( 4 \right)$
Substitute the equation $\left( 4 \right)$ in the equation $\left( 2 \right)$.
$10 + 2g + 6\left( 0 \right) = 0$
$ \Rightarrow 2g = - 10$
Divide both sides by 2.
$ \Rightarrow g = - 5$ $.....\left( 5 \right)$
Now substitute the equations $\left( 1 \right)$, $\left( 4 \right)$ and $\left( 5 \right)$ in the standard equation of a circle.
${x^2} + {y^2} + 2\left( { - 5} \right)x + 2\left( 0 \right)y + 0 = 0$
$ \Rightarrow {x^2} + {y^2} - 10x = 0$
Substitute the point $\left( {k,3} \right)$ in the above equation.
${k^2} + {3^2} - 10k = 0$
$ \Rightarrow {k^2} - 10k + 9 = 0$
Factorize the quadratic equation.
${k^2} - 9k - k + 9 = 0$
$ \Rightarrow k\left( {k - 9} \right) - \left( {k - 9} \right) = 0$
$ \Rightarrow \left( {k - 9} \right)\left( {k - 1} \right) = 0$
$ \Rightarrow k - 9 = 0$ or $k - 1 = 0$
$ \Rightarrow k = 9$ or $k = 1$
Thus, the values of $k$ are: $k = 1$ or $k = 9$
Option ‘B’ is correct
Note: Concyclic points are defined as any two or more points that lie on the common circle. All concyclic points are the same distance from the center of the circle.
Formula Used:
The standard equation of a circle is: ${x^2} + {y^2} + 2gx + 2fy + c = 0$, where $\left( { - g, - f} \right)$ is the center of a circle.
Complete step by step solution:
Given: The points $\left( {0,0} \right),\left( {1,3} \right),\left( {2,4} \right)$, and $\left( {k,3} \right)$ are concyclic.
We know that if points are concyclic, then they lie on the same circle.
Substitute the point $\left( {0,0} \right)$ in the standard equation of a circle.
We get,
${0^2} + {0^2} + 2g\left( 0 \right) + 2f\left( 0 \right) + c = 0$
$ \Rightarrow c = 0$ $.....\left( 1 \right)$
Now substitute the point $\left( {1,3} \right)$ in the standard equation of a circle.
We get,
${1^2} + {3^2} + 2g\left( 1 \right) + 2f\left( 3 \right) = 0$
$ \Rightarrow 1 + 9 + 2g + 6f = 0$
$ \Rightarrow 10 + 2g + 6f = 0$ $.....\left( 2 \right)$
Now substitute the point $\left( {2,4} \right)$ in the standard equation of a circle.
We get,
${2^2} + {4^2} + 2g\left( 2 \right) + 2f\left( 4 \right) = 0$
$ \Rightarrow 4 + 16 + 4g + 8f = 0$
$ \Rightarrow 20 + 4g + 8f = 0$
Divide both sides by 2.
$10 + 2g + 4f = 0$
$ \Rightarrow 10 + 2g = - 4f$ $.....\left( 3 \right)$
Substitute the equation $\left( 3 \right)$ in the equation $\left( 2 \right)$.
$ - 4f + 6f = 0$
$ \Rightarrow 2f = 0$
$ \Rightarrow f = 0$ $.....\left( 4 \right)$
Substitute the equation $\left( 4 \right)$ in the equation $\left( 2 \right)$.
$10 + 2g + 6\left( 0 \right) = 0$
$ \Rightarrow 2g = - 10$
Divide both sides by 2.
$ \Rightarrow g = - 5$ $.....\left( 5 \right)$
Now substitute the equations $\left( 1 \right)$, $\left( 4 \right)$ and $\left( 5 \right)$ in the standard equation of a circle.
${x^2} + {y^2} + 2\left( { - 5} \right)x + 2\left( 0 \right)y + 0 = 0$
$ \Rightarrow {x^2} + {y^2} - 10x = 0$
Substitute the point $\left( {k,3} \right)$ in the above equation.
${k^2} + {3^2} - 10k = 0$
$ \Rightarrow {k^2} - 10k + 9 = 0$
Factorize the quadratic equation.
${k^2} - 9k - k + 9 = 0$
$ \Rightarrow k\left( {k - 9} \right) - \left( {k - 9} \right) = 0$
$ \Rightarrow \left( {k - 9} \right)\left( {k - 1} \right) = 0$
$ \Rightarrow k - 9 = 0$ or $k - 1 = 0$
$ \Rightarrow k = 9$ or $k = 1$
Thus, the values of $k$ are: $k = 1$ or $k = 9$
Option ‘B’ is correct
Note: Concyclic points are defined as any two or more points that lie on the common circle. All concyclic points are the same distance from the center of the circle.
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