Question

A circle touches x-axis and cuts off a chord of length 2l from y-axis. The locus of the centre of the circle is.

• A. A straight line
• B. A circle
• C. An ellipse
• D. A hyperbola

Answer 1

Answer: (D)

A hyperbola

Answer 2

If the circle $x ^ { 2 } + y ^ { 2 } + 2 g x + 2 f y + c = 0$ touches the x-axis,

then $- f = \sqrt { g ^ { 2 } + f ^ { 2 } - c } \Rightarrow g ^ { 2 } = c$ .....(i)

and cuts a chord of length 2l from y-axis

$\Rightarrow 2 \sqrt { f ^ { 2 } - c } = 2 l \Rightarrow f ^ { 2 } - c = l ^ { 2 }$ ….(ii)

Subtracting (i) from (ii), we get $f ^ { 2 } - g ^ { 2 } = l ^ { 2 }$ .

Hence the locus is $y ^ { 2 } - x ^ { 2 } = l ^ { 2 }$, which is obviously a hyperbola.