Question

The centres of those circles which touches the circle, $x^2+y^2-8x-8y-4=0$ externally and also touch the $x$-axis, lie on:

• A. a circle
• B. an ellipse which is not a circle
• C. a hyperbola
• D. a parabola

Answer 1

Answer: (D)

a parabola

Let $C_1\Rightarrow x^2+y^2-8x-8y-4=0$

Let a circle $C_2\Rightarrow (x-h{)}^2+(y-k{)}^2=r^2$ touch the given circle

$\therefore r_1=\sqrt{g^2+f^2-c}$

$=\sqrt{16+16+4}=\sqrt{36}=6$

$r_1=6$

$r_2=r$

$(h-4{)}^2+(k-4{)}^2=(6+r{)}^2$

$C_2$ touches x-axis, so $k=r$

$(h-4{)}^2+(k-4{)}^2=(6+k{)}^2$

$h^2+16-8h+k^2+16-8k=36+k^2+12k$

$(h-4{)}^2-20k-4=0$

$(h-4{)}^2=20k+20$

Replacing $(h,k)$ by $(x,y)$

$(x-4{)}^2=4(5y+5)$

So, locus is a parabola.