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

Byju's Answer

Standard IX

Mathematics

Angle Subtended by an Arc of a Circle on the Circle and at the Center

3 circle of r...

Question

$3$ circle of radii $1, 2$ and $3$ and centres at $A, B, C$ respectively, touch each other. Another circle whose centre is $P$ touches all these $3$ circles externally and has radius $r$ . Also $∠ P A B = θ$ and $∠ P A C = α$

cos θ = \frac{3 - r}{3 (1 + r)}

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

cos α = \frac{2 - r}{2 (1 + r)}

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

r = \frac{6}{23}

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

r = \frac{6}{\sqrt{23}}

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

Open in App

Solution

The correct options are
A $cos θ = \frac{3 - r}{3 (1 + r)}$
B $cos α = \frac{2 - r}{2 (1 + r)}$
C $r = \frac{6}{23}$
$△ A B C$ is right angle
Applying cosine rule in $△ P A B$
$cos θ = \frac{3^{2} + {(1 + r)}^{2} - {(2 + r)}^{2}}{2.3 (1 + r)}$
$= \frac{3 - r}{3 (1 + r)}$
Again applying cosine rule in $△ P A C$
$cos α = \frac{{(1 + r)}^{2} + 4^{2} - {(3 + r)}^{2}}{2.4 (1 + r)} = \frac{2 - r}{2 (1 + r)}$
$∵ α + θ = 90^{o} α = 90^{o} - θ \Rightarrow cos α = sin θ$
${(\frac{3 - r}{3 (r + 1)})}^{2} + {(\frac{2 - r}{2 (r + 1)})}^{2} = 1$

Suggest Corrections

Similar questions

Q. Three circles

C_{1}, C_{2}

and

C_{3}

with radii

r_{1}, r_{2}

and

r_{1} + r_{2}

respectively are such that

C_{1}

and

C_{2}

touch each other externally and

C_{3}

internally. Another circle with radius

r_{3}

touches all the three circles. If

r_{1} > r_{2} > r_{3}

and

r_{1}, r_{2}, r_{3}

are in A.P. then

{(\frac{r_{1}}{r_{2}})}^{3}

Q. A circle centered at

P

with radius

4 c m

another circle centered at

Q

with radius

16 c m

touch each other externally. A third circle with centre

R

is drawn to touch the first two circles and one of thier common tangents as shown in the figure. Then, the radius of the circle with centre

R

The circle $C_{1} : x^{2} + y^{2} = 3$ having centre at origin,O intersects the parabola $x^{2} = 2 y$ at the point P in the first quadrant. Let the tangent to the circle $C_{1}$ at P touches other two circles $C_{2}$ and $C_{3}$ at $R_{2}$ and $R_{3}$ , respectively. Suppose $C_{2}$ and $C_{3}$ have equal radii $2 \sqrt{3}$ and centres $Q_{2}$ and $Q_{3}$ , respectively. If $Q_{2}$ and $Q_{3}$ lie on the Y-axis then

Q. The circle

C_{1} : x^{2} + y^{2} = 3

, with centre at

O

, intersect the parabola

x^{2} = 2 y

at the point

P

in the first quadrant. Let the tangent to the circle

C_{1}

P

touches other two circles

C_{2}

and

C_{3}

R_{2}

and

R_{3}

, respectively. Suppose

C_{2}

and

C_{3}

have equal radii

(2 \sqrt{3})

and centres

Q_{2}

and

Q_{3}

respectively. If

Q_{2}

and

Q_{3}

lie on the y-axis, then:

Three circle touch each other externally. A triangle is formed when the centres of these circles are joined together. Find the radii of the circles, if the sides of the triangle formed are 6 cm, 8 cm and 9 cm

Join BYJU'S Learning Program

3 circle of radii 1,2 and 3 and centres at A,B,C respectively, touch each other. Another circle whose centre is P touches all these 3 circles externally and has radius r. Also ∠PAB=θ and ∠PAC=α

$3$ circle of radii $1, 2$ and $3$ and centres at $A, B, C$ respectively, touch each other. Another circle whose centre is $P$ touches all these $3$ circles externally and has radius $r$ . Also $∠ P A B = θ$ and $∠ P A C = α$