Question

Locus of the centre of the circle which touches the two circles $x^2+y^2+8x-9=0$ and $x^2+y^2-8x+7=0$ externally, is a conic whose

• A. eccentricity is 2
• B. auxiliary circle is $x^2+y^2=1$
• C. length of latus rectum is 30
• D. co-ordinates of focus are $(4,0)$ and $(-4,0)$

Answer 1

Answer: (B), (C), (D)

As, $\left|c{c}_1-c{c}_2\right|=\left|\left(r+r_1\right)-\left(r+r_2\right)\right|=$ constant
where $\left|r_1-r_2\right|<c_1{c}_2$
$\Rightarrow$ locus of $C$ is a hyperbola with foci $c_1$ and $c_2$ i.e., $(-4,0)$ and $(4,0)$.
Also, $2a=\left|r_1-r_2\right|=2\Rightarrow a=1$
Now, $e=\frac{2ae}{2a}=\frac{8}{2}=4$
So, $b^2=1^2\left(4^2-1\right)=15$
Hence, locus of centre of circle is hyperbola, whose equation is $\frac{x^2}{1}-\frac{y^2}{15}=1$.