Find the equation of the circle passing through the intersection of the lines \[3x+y=4\] and \[x-3y+2=0\] and concentric with the circle \[2\left( {{x}^{2}}+{{y}^{2}} \right)-3x+8y-1=0\].
Answer
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Hint: Two concentric circles have same center and distance from center to any point on circumference is radius
Given lines, \[{{L}_{1}}=3x+y=4....\left( i \right)\]
\[{{L}_{2}}=x-3y+2=0....\left( ii \right)\]
We have to find the point of intersection of \[{{ L }_{1}}\]and \[{{L}_{2}}\], we will solve equation \[\left( i \right)\]and \[\left( ii \right)\]together.
Taking equation\[\left( i \right)\], \[3x+y=4\]
We get, \[y=4-3x....\left( iii \right)\]
Putting value of \[y\] in equation \[\left( ii \right)\]
\[x-3y+2=0\]
\[x-3\left[ 4-3x \right]+2=0\]
\[x-12+9x+2=0\]
\[10x=10\]
Therefore, we get
Putting value of \[x\] in equation \[\left( iii \right)\]to find the value of \[y\]
\[y=4-3x\]
\[=4-3\left( 1 \right)\]
We get \[y=1\]
Therefore circle \[{{C}_{2}}\]passes through\[\left( x,y \right)=\left( 1,1 \right)\].
Now circle \[{{C}_{2}}\]is concentric with circle:
\[{{C}_{1}}=2\left( {{x}^{2}}+{{y}^{2}} \right)-3x+8y-1=0\]
Dividing the equation by \[2\],
We get \[{{C}_{1}}=\left( {{x}^{2}}+{{y}^{2}} \right)-\dfrac{3x}{2}+4y-\dfrac{1}{2}=0....\left( iv \right)\]
The general equation of circle is,
\[{{x}^{2}}+{{y}^{2}}+2gx+2fy+c=0\]
Now, we will compare general equation of circle with equation\[\left( iv \right)\].
We get, \[2g=-\dfrac{3}{2}\]
\[g=\dfrac{-3}{4}\]
\[2f=4\]
\[f=2\]
We know that center of circle is \[\left( -g,-f \right)\].
Therefore, \[\text{centre}=\left( \dfrac{3}{4},-2 \right)\]
Since, given circle is concentric with circle to be found \[\left( {{C}_{2}} \right)\], both would have same center.
Now, the new circle \[{{C}_{2}}\]has center at \[C\left( \dfrac{3}{4},-2 \right)\]and it passes through intersection of line \[A=\left( 1,1 \right)\].
Distance of point \[A\left( 1,1 \right)\]to \[C\left( \dfrac{3}{4},-2 \right)\]is radius.
By distance formula,
\[r=AC=\sqrt{{{\left( {{x}_{2}}-{{x}_{1}} \right)}^{2}}+{{\left( {{y}_{2}}-{{y}_{1}} \right)}^{2}}}\]
\[r=AC=\sqrt{{{\left( 1-\dfrac{3}{4} \right)}^{2}}+{{\left( 1+2 \right)}^{2}}}\]
\[=\sqrt{\dfrac{1}{16}+9}\]
We get radius \[=\sqrt{\dfrac{145}{16}}=\dfrac{\sqrt{145}}{4}\text{ units}\]
Now we know that, equation of circle with center \[\left( a,b \right)\]and radius \[r\]
\[={{\left( x-a \right)}^{2}}+{{\left( y-b \right)}^{2}}={{r}^{2}}\]
Here, we found the center \[\left( \dfrac{3}{4},-2 \right)\]and radius\[=\dfrac{\sqrt{145}}{4}\text{ units}\]
We get, equation of circle \[={{\left( x-\dfrac{3}{4} \right)}^{2}}+{{\left[ y-\left( -2 \right) \right]}^{2}}=\dfrac{145}{16}\]
\[=16{{x}^{2}}+16{{y}^{2}}-24x+64y=72\]
Therefore, final equation of circle is:
\[2\left( {{x}^{2}}+{{y}^{2}} \right)-3x+8y=9\]
Note:
Elimination method is also suitable to find the intersection of line. Students can also use the method of family of circles passing through a point and having a center with constraint.
Given lines, \[{{L}_{1}}=3x+y=4....\left( i \right)\]
\[{{L}_{2}}=x-3y+2=0....\left( ii \right)\]
We have to find the point of intersection of \[{{ L }_{1}}\]and \[{{L}_{2}}\], we will solve equation \[\left( i \right)\]and \[\left( ii \right)\]together.
Taking equation\[\left( i \right)\], \[3x+y=4\]
We get, \[y=4-3x....\left( iii \right)\]
Putting value of \[y\] in equation \[\left( ii \right)\]
\[x-3y+2=0\]
\[x-3\left[ 4-3x \right]+2=0\]
\[x-12+9x+2=0\]
\[10x=10\]
Therefore, we get
Putting value of \[x\] in equation \[\left( iii \right)\]to find the value of \[y\]
\[y=4-3x\]
\[=4-3\left( 1 \right)\]
We get \[y=1\]
Therefore circle \[{{C}_{2}}\]passes through\[\left( x,y \right)=\left( 1,1 \right)\].
Now circle \[{{C}_{2}}\]is concentric with circle:
\[{{C}_{1}}=2\left( {{x}^{2}}+{{y}^{2}} \right)-3x+8y-1=0\]
Dividing the equation by \[2\],
We get \[{{C}_{1}}=\left( {{x}^{2}}+{{y}^{2}} \right)-\dfrac{3x}{2}+4y-\dfrac{1}{2}=0....\left( iv \right)\]
The general equation of circle is,
\[{{x}^{2}}+{{y}^{2}}+2gx+2fy+c=0\]
Now, we will compare general equation of circle with equation\[\left( iv \right)\].
We get, \[2g=-\dfrac{3}{2}\]
\[g=\dfrac{-3}{4}\]
\[2f=4\]
\[f=2\]
We know that center of circle is \[\left( -g,-f \right)\].
Therefore, \[\text{centre}=\left( \dfrac{3}{4},-2 \right)\]
Since, given circle is concentric with circle to be found \[\left( {{C}_{2}} \right)\], both would have same center.
Now, the new circle \[{{C}_{2}}\]has center at \[C\left( \dfrac{3}{4},-2 \right)\]and it passes through intersection of line \[A=\left( 1,1 \right)\].
Distance of point \[A\left( 1,1 \right)\]to \[C\left( \dfrac{3}{4},-2 \right)\]is radius.
By distance formula,
\[r=AC=\sqrt{{{\left( {{x}_{2}}-{{x}_{1}} \right)}^{2}}+{{\left( {{y}_{2}}-{{y}_{1}} \right)}^{2}}}\]
\[r=AC=\sqrt{{{\left( 1-\dfrac{3}{4} \right)}^{2}}+{{\left( 1+2 \right)}^{2}}}\]
\[=\sqrt{\dfrac{1}{16}+9}\]
We get radius \[=\sqrt{\dfrac{145}{16}}=\dfrac{\sqrt{145}}{4}\text{ units}\]
Now we know that, equation of circle with center \[\left( a,b \right)\]and radius \[r\]
\[={{\left( x-a \right)}^{2}}+{{\left( y-b \right)}^{2}}={{r}^{2}}\]
Here, we found the center \[\left( \dfrac{3}{4},-2 \right)\]and radius\[=\dfrac{\sqrt{145}}{4}\text{ units}\]
We get, equation of circle \[={{\left( x-\dfrac{3}{4} \right)}^{2}}+{{\left[ y-\left( -2 \right) \right]}^{2}}=\dfrac{145}{16}\]
\[=16{{x}^{2}}+16{{y}^{2}}-24x+64y=72\]
Therefore, final equation of circle is:
\[2\left( {{x}^{2}}+{{y}^{2}} \right)-3x+8y=9\]
Note:
Elimination method is also suitable to find the intersection of line. Students can also use the method of family of circles passing through a point and having a center with constraint.
Last updated date: 19th Sep 2023
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