The locus of centre of a variable circle touching the circles of radius r1 and r2 externally, which also touch each other externally, is a conic of the eccentricity e. If (r1/r2)=3+2 √2, then e2 is

Question

The locus of centre of a variable circle touching the circles of radius r1 and r2 externally, which also touch each other externally, is a conic of the eccentricity e. If (r1/r2)=3+2 √2, then e2 is (A) 2 (B) 3 (C) 4 (D) 5

Tardigrade Admin · Accepted Answer

<img class="img-fluid question-image" alt="image" src="https://cdn.tardigrade.in/img/question/mathematics/2bebb909fd5bf73bedeeb4ddff3f4b0e-.png" /> cc 2= r + r 2 cc 1- cc 2= r 1- r 2 ⇒ Locus of c is hyperbola with foci c 1 and c 2 ⇒ 2 ae = c 1 c 2= r 1+ r 2 and r 1- r 2=2 a ⇒ ( r 1+ r 2/ r 1- r 2)=√2 ⇒ e 2=2

Q. The locus of centre of a variable circle touching the circles of radius $r_{1}$ and $r_{2}$ externally, which also touch each other externally, is a conic of the eccentricity e. If $\frac{r _{1}}{r _{2}} = 3 + 22$ , then $e^{2}$ is

2

3

4

5

$c c_{2} = r + r_{2}$
$c c_{1} - c c_{2} = r_{1} - r_{2}$
$\Rightarrow$ Locus of $c$ is hyperbola with foci $c_{1}$ and $c_{2}$
$\Rightarrow 2 a e = c_{1} c_{2} = r_{1} + r_{2}$ and $r_{1} - r_{2} = 2 a$
$\Rightarrow \frac{r _{1} + r _{2}}{r _{1} - r _{2}} = 2$
$\Rightarrow e^{2} = 2$

Q. The locus of centre of a variable circle touching the circles of radius r1​ and r2​ externally, which also touch each other externally, is a conic of the eccentricity e. If r2​r1​​=3+22​, then e2 is

2

3

4

5

cc2​=r+r2​ cc1​−cc2​=r1​−r2​ ⇒ Locus of c is hyperbola with foci c1​ and c2​ ⇒2ae=c1​c2​=r1​+r2​ and r1​−r2​=2a ⇒r1​−r2​r1​+r2​​=2​ ⇒e2=2

Q. The locus of centre of a variable circle touching the circles of radius $r_{1}$ and $r_{2}$ externally, which also touch each other externally, is a conic of the eccentricity e. If $\frac{r _{1}}{r _{2}} = 3 + 22$ , then $e^{2}$ is

$c c_{2} = r + r_{2}$
$c c_{1} - c c_{2} = r_{1} - r_{2}$
$\Rightarrow$ Locus of $c$ is hyperbola with foci $c_{1}$ and $c_{2}$
$\Rightarrow 2 a e = c_{1} c_{2} = r_{1} + r_{2}$ and $r_{1} - r_{2} = 2 a$
$\Rightarrow \frac{r _{1} + r _{2}}{r _{1} - r _{2}} = 2$
$\Rightarrow e^{2} = 2$