If the chord of contact of the tangents from the point (α , β ) to the circle x2+y2=r12 is a tangent to the circle (x - a)2+(y - b)2=r22 , then

Question

If the chord of contact of the tangents from the point (α , β ) to the circle x2+y2=r12 is a tangent to the circle (x - a)2+(y - b)2=r22 , then (A) r22(α2+β2)=(r12-a α-b β)2 (B) r22(α2+β2)=(r12+a α-b β)2 (C) r22(α2+β2)=(r12-a α+b β)2 (D) r22(α2+β2)=(r12+a α+b β)2

Tardigrade Admin · Accepted Answer

The equation of the chord of contact of tangents drawn from a point

(α, β)

to the circle

x^{2} + y^{2} = r_{1}^{2}

is

αx + β y = r_{1}^{2}

This line touches the circle

(x - a)^{2} + (y - b)^{2} = r_{2}^{2}

∴ ∣ ∣ \frac{a α + b β - r _{1}^{2}}{α ^{2} + β ^{2}} ∣ ∣ = r_{2}

\Rightarrow r_{2}^{2} (α^{2} + β^{2}) = (a α + b β - r_{1}^{2})^{2}

\Rightarrow r_{2}^{2} (α^{2} + β^{2}) = (r_{1}^{2} - a α - b β)^{2}

Q. If the chord of contact of the tangents from the point $(α, β)$ to the circle $x^{2} + y^{2} = r_{1}^{2}$ is a tangent to the circle $(x - a)^{2} + (y - b)^{2} = r_{2}^{2}$ , then

$r_{2}^{2} (α^{2} + β^{2}) = (r_{1}^{2} - a α - b β)^{2}$

$r_{2}^{2} (α^{2} + β^{2}) = (r_{1}^{2} + a α - b β)^{2}$

$r_{2}^{2} (α^{2} + β^{2}) = (r_{1}^{2} - a α + b β)^{2}$

$r_{2}^{2} (α^{2} + β^{2}) = (r_{1}^{2} + a α + b β)^{2}$

Q. If the chord of contact of the tangents from the point (α,β) to the circle x2+y2=r12​ is a tangent to the circle (x−a)2+(y−b)2=r22​ , then

r22​(α2+β2)=(r12​−aα−bβ)2

r22​(α2+β2)=(r12​+aα−bβ)2

r22​(α2+β2)=(r12​−aα+bβ)2

r22​(α2+β2)=(r12​+aα+bβ)2

Q. If the chord of contact of the tangents from the point $(α, β)$ to the circle $x^{2} + y^{2} = r_{1}^{2}$ is a tangent to the circle $(x - a)^{2} + (y - b)^{2} = r_{2}^{2}$ , then

$r_{2}^{2} (α^{2} + β^{2}) = (r_{1}^{2} - a α - b β)^{2}$

$r_{2}^{2} (α^{2} + β^{2}) = (r_{1}^{2} + a α - b β)^{2}$

$r_{2}^{2} (α^{2} + β^{2}) = (r_{1}^{2} - a α + b β)^{2}$

$r_{2}^{2} (α^{2} + β^{2}) = (r_{1}^{2} + a α + b β)^{2}$