Answer

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Hint: In order to solve this equation, we have to find out the value of c by using a circle equation that passes through the point (2,0), (0,1), (4,5), (0,c).

Complete step-by-step answer:

Concyclic points – A set of points are said to be concyclic, if they lie on a common circle. All concyclic points are the same distance from the centre of the circle.

By using Circle equation we get , ${x^2} + {y^2} + 2gx + 2fy + c = 0$

All the points (2,0), (0,1), (4,5), (0,c) will lie on this circle

For point A (2,0) -

${\left( 2 \right)^2} + {\left( 0 \right)^2} + 2 \times g \times 2 + c = 0$

$4 + 4g + c = 0$ ….. (1)

For point B (0,1)

${\left( 1 \right)^2} + {\text{ 2f}} \times {\text{1 + }}c{\text{ = 0}}$

$2f{\text{ + c + 1 = 0}}$ ….. (2)

For point (4,5)

$

{\left( 4 \right)^2} + {\left( 5 \right)^2} + 2g \times 4 + 2f \times 5 + c = 0 \\

41{\text{ }} + 8g{\text{ }} + 10f + c = 0{\text{ }}.....{\text{(3)}} \\

$

By solving these equations we get,

From $(1)$

$

{\text{ }}4 + 4g + c = 0 \\

c = - 4 - 4g \\

{\text{ }} \\

$

Put the value of c in equation $(3)$

$

{\text{ }}41 + 8g + 10f - 4 - 4g = 0 \\

\Rightarrow {\text{ 37 + 4g + 10f = 0 }}....{\text{ (4)}} \\

$

Again, put the value of ‘c’ from equation $(1)$, in equation $(2)$, we get,

$

{\text{ }}1 + 2f + c = 0 \\

\Rightarrow 1 + 2f - 4 - 4g = 0 \\

\Rightarrow 2f - 4g - 3 = 0{\text{ }}.....{\text{(5) }} \\

$

From equation, $(4)$ and $(5)$, we get,

$

37 + 4g + 10f = 0 \\

\& {\text{ 2f - 4g - 3 = 0}} \\

$

$

\Rightarrow 4g + 10f = - 37{\text{ }}...{\text{(A)}} \\

\Rightarrow {\text{2}}f - 4g = 3{\text{ }}....{\text{ (B)}} \\

$

From equation $(A)$, we get

$

\Rightarrow 4g = - 37 - 10f \\

\Rightarrow g = \dfrac{{ - 37 - 10f}}{4} \\

\Rightarrow g = \dfrac{{ - 1}}{4}\left( {37 - 10f} \right){\text{ }} \to {\text{(C)}} \\

$

Put the value of $g$ from equation $(C)$ in $(B)$, we get

$

{\text{ }}2f - 4\left( {\dfrac{{ - 1}}{4}\left( {37,10f} \right)} \right){\text{ = 3}} \\

\Rightarrow 2f + 37 + 10f = 3 \\

\Rightarrow 12f + 37 = 3 \\

\Rightarrow 12f = - 34 \\

\Rightarrow f = - \dfrac{{17}}{6} \\

$

Put the value of $f$ in equation $(C)$, we get

$g = - \dfrac{1}{4}\left( {37 + 10f} \right)$

$

\Rightarrow {\text{ g = - }}\dfrac{1}{4}\left( {37 + 10\left( { - \dfrac{{17}}{6}} \right)} \right) \\

\Rightarrow {\text{ g = - }}\dfrac{1}{4}\left( {37 - \dfrac{{170}}{6}} \right) \\

\Rightarrow {\text{ g = - }}\dfrac{1}{4}\left( {\dfrac{{222 - 170}}{6}} \right) \\

\Rightarrow {\text{ g = - }}\dfrac{1}{4}\left( {\dfrac{{52}}{6}} \right) \\

$

$ \Rightarrow {\text{ g = - }}\dfrac{{13}}{6}$

Put the value of $g$ in equation $(1)$

$

{\text{ }}4 + 4g + c = 0 \\

\Rightarrow 4 + 4\left( { - \dfrac{{13}}{6}} \right) + c = 0 \\

\Rightarrow 4 - \dfrac{{52}}{6}{\text{ + c = 0}} \\

\Rightarrow \dfrac{{24 - 52}}{6}{\text{ + c = 0}} \\

$

$

\Rightarrow - \dfrac{{28}}{6} + c = 0 \\

\Rightarrow c = \dfrac{{14}}{3} \\

$

$\therefore $ The correct answer is option C.

Note: Whenever we face such types of questions, the key concept is that we have to write what is given to us, like we did. Then we will apply the circle equation to calculate concyclic points, after deriving all the equations and solving them, we find the value of c and thus we get our answer.

Complete step-by-step answer:

Concyclic points – A set of points are said to be concyclic, if they lie on a common circle. All concyclic points are the same distance from the centre of the circle.

By using Circle equation we get , ${x^2} + {y^2} + 2gx + 2fy + c = 0$

All the points (2,0), (0,1), (4,5), (0,c) will lie on this circle

For point A (2,0) -

${\left( 2 \right)^2} + {\left( 0 \right)^2} + 2 \times g \times 2 + c = 0$

$4 + 4g + c = 0$ ….. (1)

For point B (0,1)

${\left( 1 \right)^2} + {\text{ 2f}} \times {\text{1 + }}c{\text{ = 0}}$

$2f{\text{ + c + 1 = 0}}$ ….. (2)

For point (4,5)

$

{\left( 4 \right)^2} + {\left( 5 \right)^2} + 2g \times 4 + 2f \times 5 + c = 0 \\

41{\text{ }} + 8g{\text{ }} + 10f + c = 0{\text{ }}.....{\text{(3)}} \\

$

By solving these equations we get,

From $(1)$

$

{\text{ }}4 + 4g + c = 0 \\

c = - 4 - 4g \\

{\text{ }} \\

$

Put the value of c in equation $(3)$

$

{\text{ }}41 + 8g + 10f - 4 - 4g = 0 \\

\Rightarrow {\text{ 37 + 4g + 10f = 0 }}....{\text{ (4)}} \\

$

Again, put the value of ‘c’ from equation $(1)$, in equation $(2)$, we get,

$

{\text{ }}1 + 2f + c = 0 \\

\Rightarrow 1 + 2f - 4 - 4g = 0 \\

\Rightarrow 2f - 4g - 3 = 0{\text{ }}.....{\text{(5) }} \\

$

From equation, $(4)$ and $(5)$, we get,

$

37 + 4g + 10f = 0 \\

\& {\text{ 2f - 4g - 3 = 0}} \\

$

$

\Rightarrow 4g + 10f = - 37{\text{ }}...{\text{(A)}} \\

\Rightarrow {\text{2}}f - 4g = 3{\text{ }}....{\text{ (B)}} \\

$

From equation $(A)$, we get

$

\Rightarrow 4g = - 37 - 10f \\

\Rightarrow g = \dfrac{{ - 37 - 10f}}{4} \\

\Rightarrow g = \dfrac{{ - 1}}{4}\left( {37 - 10f} \right){\text{ }} \to {\text{(C)}} \\

$

Put the value of $g$ from equation $(C)$ in $(B)$, we get

$

{\text{ }}2f - 4\left( {\dfrac{{ - 1}}{4}\left( {37,10f} \right)} \right){\text{ = 3}} \\

\Rightarrow 2f + 37 + 10f = 3 \\

\Rightarrow 12f + 37 = 3 \\

\Rightarrow 12f = - 34 \\

\Rightarrow f = - \dfrac{{17}}{6} \\

$

Put the value of $f$ in equation $(C)$, we get

$g = - \dfrac{1}{4}\left( {37 + 10f} \right)$

$

\Rightarrow {\text{ g = - }}\dfrac{1}{4}\left( {37 + 10\left( { - \dfrac{{17}}{6}} \right)} \right) \\

\Rightarrow {\text{ g = - }}\dfrac{1}{4}\left( {37 - \dfrac{{170}}{6}} \right) \\

\Rightarrow {\text{ g = - }}\dfrac{1}{4}\left( {\dfrac{{222 - 170}}{6}} \right) \\

\Rightarrow {\text{ g = - }}\dfrac{1}{4}\left( {\dfrac{{52}}{6}} \right) \\

$

$ \Rightarrow {\text{ g = - }}\dfrac{{13}}{6}$

Put the value of $g$ in equation $(1)$

$

{\text{ }}4 + 4g + c = 0 \\

\Rightarrow 4 + 4\left( { - \dfrac{{13}}{6}} \right) + c = 0 \\

\Rightarrow 4 - \dfrac{{52}}{6}{\text{ + c = 0}} \\

\Rightarrow \dfrac{{24 - 52}}{6}{\text{ + c = 0}} \\

$

$

\Rightarrow - \dfrac{{28}}{6} + c = 0 \\

\Rightarrow c = \dfrac{{14}}{3} \\

$

$\therefore $ The correct answer is option C.

Note: Whenever we face such types of questions, the key concept is that we have to write what is given to us, like we did. Then we will apply the circle equation to calculate concyclic points, after deriving all the equations and solving them, we find the value of c and thus we get our answer.

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