વર્તુળ x^2 + y^2 - 4x - 6y - 21 = 0 અને | vedclass

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Question

Point $(1,2)$ lies on the circle ${x^2} + {y^2} + 2x + 2y - 11 = 0\,\,\,$, because coordinates of point $(1,2)$ satisfy the equation

${x^2} +

Vedclass · Accepted Answer

$x^2 +y^2 + 2x + 2y - 11 = 0$

Vedclass · Answer

Point $(1,2)$ lies on the circle ${x^2} + {y^2} + 2x + 2y - 11 = 0\,\,\,$, because coordinates of point $(1,2)$ satisfy the equation

${x^2} + {y^2} + 2x + 2y - 11 = 0\,\,\,$

Now, ${x^2} + {y^2} - 4x + 6y - 21 = 0\,\,\,\,\,\,\,\,\,........\left( i ight)$

${x^2} + {y^2} + 2x + 2y - 11 = 0\,\,\,\,\,\,\,\,\,..........\left( {ii} ight)$

$3x + 4y + 5 = 0\,\,\,\,\,\,\,\,\,..........\left( {iii} ight)$

From $(i)$ and $(iii)$

${x^2} + {\left( { - \frac{{3x + 5}}{4}} ight)^2} - 4x - 6{\left( { - \frac{{3x + 5}}{4}} ight)^2} - 21 = 0$

$\begin{array}{l}
 \Rightarrow 16{x^2} + 9{x^2} + 30x + 25 - 64x\
\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, + 72x + 120 - 336 = 0
\end{array}$

$ \Rightarrow 25{x^2} + 38x - 191 = 0\,\,\,\,\,\,\,\,\,\,.......\left( {iv} ight)$

From $(ii)$ abd $(iii)$,

${x^2} + {\left( { - \frac{{3x + 5}}{4}} ight)^2} + 2x + 2{\left( { - \frac{{3x + 5}}{4}} ight)^2} - 11 = 0$

$\begin{array}{l}
 \Rightarrow 16{x^2} + 9{x^2} + 30x + 25 + 32x\
\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, - 24x - 40 - 176 = 0
\end{array}$

$ \Rightarrow 25{x^2} + 38x - 191 = 0\,\,\,\,\,\,\,\,\,\,.......\left( v ight)$

Thus we get the same equation from $(ii)$ and $(iii)$ as we get from equation $(i)$ and $(iii)$. Hence the point of intersction of $(ii)$ and $(iii)$ will be sameas the point of intersections of $(i)$ and $(iii)$, Therefore the circle $(ii)$ passing through the point of  inersection of circle $(i)$ and poin $(1,2)$ also as show in the figure.

Hence equation $(ii)$ i.e

${x^2} + {y^2} + 2x + 2y - 11 = 0\,\,\,$ is the equation of required circle.

વર્તુળ $x^2 + y^2 - 4x - 6y - 21 = 0$ અને $3x + 4y + 5 = 0$ ના છેદબિંદુ અને બિંદુ $(1,2)$ માંથી પસાર થતાં વર્તુળનું સમીકરણ મેળવો.

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