JEE Advanced · 2013 · 02 Jun 2013 · Paper 2 · Circle

Circle(s) touching

x

-axis at a distance 3 from the origin and having an intercept of length

27

y

-axis is (are)

Answer: (A), (C)

Solution

A circle touching the

x

-axis at distance 3 from origin has center at

(3, k)

(- 3, k)

where

k

is the radius.

For center at

(3, k)

: equation is

(x - 3)^{2} + (y - k)^{2} = k^{2}

.

The

y

-intercept length is

27

. Setting

x = 0

9 + (y - k)^{2} = k^{2}

(y - k)^{2} = k^{2} - 9

y = k \pm k^{2} - 9

The intercept length is

2 k^{2} - 9 = 27

, so

k^{2} - 9 = 7

, giving

k^{2} = 16

, thus

k = \pm 4

.

For

k = 4

(x - 3)^{2} + (y - 4)^{2} = 16

expands to

x^{2} + y^{2} - 6 x - 8 y + 9 = 0

(option C).

For

k = - 4

(x - 3)^{2} + (y + 4)^{2} = 16

expands to

x^{2} + y^{2} - 6 x + 8 y + 9 = 0

(option A).

Therefore, the answers are (A) and (C).

JEE Advanced · 2013 · 02 Jun 2013 · Paper 2 · One or more correct options