You visited us 1 times! Enjoying our articles? Unlock Full Access!

Byju's Answer

Standard XII

Mathematics

Parametric Form of Tangent: Ellipse

Circles C1 an...

Question

Circles $C_{1}$ and $C_{2}$ , of radii r and R respectively, touch each other as shown in figure. The line l, which is parallel to the line joining the centres of $C_{1}$ and $C_{2}$ , is tangent to $C_{1}$ at P and intersects $C_{2}$ at A,B. If $R^{2} = 2 r^{2}$ , then $∠ A O B$

No worries! We‘ve got your back. Try BYJU‘S free classes today!

Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses

No worries! We‘ve got your back. Try BYJU‘S free classes today!

Open in App

Solution

The correct option is B

$C_{1} (r, 0) C_{2} (R, 0) E q . o f A B y = r E q . o f c i r c l e C_{2} (x - R)^{2} + y^{2} = R^{2} A (R - \sqrt{R^{2} - r^{2}}, r) u s i n g R^{2} = 2 r^{2} B (R + \sqrt{R^{2} - r^{2}}, r) A (R - r, r), B (R + r, r) S l o p e o f O A = \frac{r}{R - r} = m_{1} S l o p e o f O B = \frac{r}{R + r} = m_{2} t a n θ = \frac{m_{1} - m_{2}}{1 + m_{1} m_{2}} = 1 θ = 45^{\circ}$

Suggest Corrections

Similar questions

Q. Circles

C_{1}

and

C_{2}

, of radii

r

and

R

respectively, touch each other as shown in figure. The line

l

, which is parallel to the line joining the centres of

C_{1}

and

C_{2}

, is tangent to

C_{1}

P

and intersects

C_{2}

A, B .

R^{2} = 2 r^{2}

, then

∠ A O B

Q. Circles

C_{1}

and

C_{2}

, of radii

r

and

R

respectively, touch each other as shown in the figure. The line

A

, which is parallel to the line joining the centres of

C_{1}

and

C_{2}

, is tangent to

C_{1}

P

and intersects

C_{2}

A, B

. If

R^{2} = 2 r^{2}

, then

∠ A O B

equals

Q. Circles

C_{1}

and

C_{2}

, of radii

r

and

R

respectively, touch each other as shown in the figure. The line

A

, which is parallel to the line joining the centres of

C_{1}

and

C_{2}

, is tangent to

C_{1}

P

and intersects

C_{2}

A, B

. If

R^{2} = 2 r^{2}

, then

∠ A O B

equals

(correct answer + 2, wrong answer - 0.50)

Q. The centres of two circles

C_{1}

and

C_{2}

each of unit radius are at a distance of 6 units from each other. Let

P

be the mid point of the line segment joining the centres of

C_{1}

and

C_{2}

and

C

be a circle touching circles

C_{1}

and

C_{2}

extemally. If a common tangent to

C_{1}

and

C

passing through

P

is also a common tangent to

C_{2}

and

C

, then the radius ofthe circle

C

Q. The centres of two circles

C_{1}

and

C_{2}

each of unit radius are at a distance of

6

units from each other. Let

P

be the mid point of the line segment joining the centres of

C_{1}

, and

C_{2}

and

C

be a circle touching

C_{1}

and

C_{2}

externally. If a common tangent to

C_{1},

and

C

passing through

P

is also a common tangent to

C_{2}

and

C

, then the radius of the circle

C

is:

Join BYJU'S Learning Program

Circles C1 and C2, of radii r and R respectively, touch each other as shown in figure. The line l, which is parallel to the line joining the centres of C1 and C2, is tangent to C1 at P and intersects C2 at A,B. If R2=2r2, then ∠AOB

The correct option is B C1(r,0)C2(R,0)Eq.of AB y=rEq.of circle C2(x−R)2+y2=R2A(R−√R2−r2,r) using R2=2r2B(R+√R2−r2,r) A(R−r,r),B(R+r,r)Slope of OA=rR−r=m1Slope of OB=rR+r=m2tanθ=m1−m21+m1m2=1θ=45∘

Circles $C_{1}$ and $C_{2}$ , of radii r and R respectively, touch each other as shown in figure. The line l, which is parallel to the line joining the centres of $C_{1}$ and $C_{2}$ , is tangent to $C_{1}$ at P and intersects $C_{2}$ at A,B. If $R^{2} = 2 r^{2}$ , then $∠ A O B$