Let circle $C$ be the image of $x^{2}+y^{2}-2 x+4 y-4=0$ in the line $2 \mathrm{x}-3 \mathrm{y}+5=0$ and A be the point on C such that OA is parallel to x -axis and A lies on the right hand side of the centre O of C . If $\mathrm{B}(\alpha, \beta)$, with $\beta<4$, lies on C such that the length of the arc AB is $(1 / 6)^{\text {th }}$ of the perimeter of C , then $\beta-\sqrt{3} \alpha$ is equal to

Circle — JEE Main · 2025 · 24 Jan 2025 · Shift 1 Question with Solution

Let circle

C

be the image of

x^{2} + y^{2} - 2 x + 4 y - 4 = 0

in the line

2 x - 3 y + 5 = 0

and A be the point on C such that OA is parallel to x -axis and A lies on the right hand side of the centre O of C . If

B (α, β)

, with

β < 4

, lies on C such that the length of the arc AB is

(1/6)^{th}

of the perimeter of C , then

β - 3 α

is equal to

Answer: (D)

The equation

x^{2} + y^{2} - 2 x + 4 y - 4 = 0

rewrites as

(x - 1)^{2} + (y + 2)^{2} = 9

, a circle of centre

(1, - 2)

and radius

3

.

Reflecting the centre in the line

2 x - 3 y + 5 = 0

: for a point

(x_{1}, y_{1})

the image satisfies

\frac{x - x _{1}}{2} = \frac{y - y _{1}}{- 3} = \frac{- 2 ( 2 x _{1} - 3 y _{1} + 5 )}{2 ^{2} + ( - 3 ) ^{2}} .

With

(x_{1}, y_{1}) = (1, - 2)

2 x_{1} - 3 y_{1} + 5 = 2 + 6 + 5 = 13

, so the factor is

\frac{- 2 ( 13 )}{13} = - 2

x = 1 + 2 (- 2) = - 3, y = - 2 + (- 3) (- 2) = 4.

So circle

C

has centre

O = (- 3, 4)

and radius

3

(x + 3)^{2} + (y - 4)^{2} = 9

A

lies on

C

with

O A

parallel to the

x

-axis on the right of

O

, so

A = (0, 4)

, corresponding to angular position

θ = 0

if points of

C

are written as

(- 3 + 3 cos θ, 4 + 3 sin θ)

.

Arc

A B

\frac{1}{6}

of the perimeter, so the angle subtended at the centre is

\frac{1}{6} (2 π) = \frac{π}{3}

. Since

β < 4

B

lies below the horizontal through

O

, so

θ = - \frac{π}{3}

α = - 3 + 3 cos (- \frac{π}{3}) = - 3 + \frac{3}{2} = - \frac{3}{2}, β = 4 + 3 sin (- \frac{π}{3}) = 4 - \frac{3 3}{2} .

β - 3 α = 4 - \frac{3 3}{2} - 3 (- \frac{3}{2}) = 4 - \frac{3 3}{2} + \frac{3 3}{2} = 4.

JEE Main · 2025 · 24 Jan 2025 · Shift 1 · Single correct option