Problem 14

For what values of $l$ and $m$, circle $5\left(x^{2}+y^{2}\right)+b y-m=0$ belongs to the co-axial system determined by the circles $x^{2}+y^{2}+2 x+4 y-6=0$ and $2\left(x^{2}+y^{2}\right)-x=0 ?$

Short Answer

Expert verified

The values of $l$ and $m$ that make the given circle belong to the coaxial system determined by the two provided circles are $2$ and approximately $1.06$ respectively.

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TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Determine the coefficients of the two circles

The equation of the first provided circle can be rewritten as $x^{2}+y^{2}+2(-1)x+2(2)y-6=0$, thus $g_1=-1$, $f_1=2$, and $c_1=-6$. Similarly for the second circle, we rewrite it as $2x^{2}+2y^{2}-2(0.5)x+0y+0=0$, hence $g_2=0.5$, $f_2=0$, and $c_2=0$. We can therefore conclude that the equation for any circle within this coaxial system will have the form $5\left(x^{2}+y^{2}\right)+10ax+10by+c=0.$

Step 2: Determine the center and radius of the auxiliary circle

The center $-g, -f$ of the auxiliary circle for this system can be obtained by averaging the centers of the two provided circles giving us $-g=(g_1+g_2)/2$ which equals $-0.25$ and $-f=(f_1+f_2)/2$ which equals $1$. The radius $a$ of the auxiliary circle is given by $\sqrt{g^2+f^2-c}$, but since $c=0$ for the given circles, the radius $a$ is simply $\sqrt{g^2+f^2} = \sqrt{(0.25)^2+1^2} = 1.03 $ to three significant figures. Therefore, the auxiliary or Soddy's circle for the given coaxial system is $x^2+y^2+0.5x-2y=0$.

Step 3: Determine the values for $l$ and $m$

The required circle must be in the form $a\left(x^{2}+y^{2}\right)+2gx+2fy+c=0$. Comparing this with the given expression for the circle, it can be concluded that the equation for $b$ will be $b=2f$, hence $b=2(1)=2$. Similarly, $m=a^2=1.03^2=1.06$. Therefore, the values of $l$ and $m$ are $2$ and $1.06$ respectively.

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Chapter 1: Chapter 1

For what values of \(l\) and \(m\), circle \(5\left(x^{2}+y^{2}\right)+b y-m=0\) belongs to the co-axial system determined by the circles $x^{2}+y^{2}+2 x+4 y-6=0\( and \)2\left(x^{2}+y^{2}\right)-x=0 ?$

Short Answer

Step by step solution

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Step 1: Determine the coefficients of the two circles

Step 2: Determine the center and radius of the auxiliary circle

Step 3: Determine the values for \(l\) and \(m\)

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More chapters from the book ‘Skills in Mathematics for All Engineering Entrance Examinations: Coordinate Geometry’

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Chapter 3

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