###### Advertisements

###### Advertisements

Points A(–1, *y*) and B(5, 7) lie on a circle with centre O(2, –3*y*). Find the values of* y*. Hence find the radius of the circle.

###### Advertisements

#### Solution

A and B are the two points that lie on the circle and O is the centre of the circle.

Therefore, OA and OB are the radii of the circle.

Using the distance formula, we have:

`OA=sqrt((-1-2)^2+(y+3y)^2)=sqrt(9+16y^2)`

`OB=sqrt((5-2)^2+(7+3y)^2)=sqrt(9+(7+3y)^2)`

Now, OB = OA (Radii of the same circle)

`sqrt(9+(7+3y)^2)=sqrt(9+16y^2)`

9+(7+3y)^{2}=9+16y^^{2} (squaring both the sides)

49+9y^{2}+42y=16y^{2}

⇒7y^{2}−42y−49=0

⇒y^{2}−6y−7 =0

⇒y^{2}−7y+y−7=0

⇒(y−7)(y+1)=0

⇒y−7=0 or y+1=0

⇒y=7 or y=−1

When y = 7:

Radius of the circle, `OA=sqrt(9+16y^2)=sqrt(9+16×49)=sqrt(793)`

When y = −1:

Radius of the circle, `OA=sqrt(9+16y^2) =sqrt(9+16×1)=sqrt(25)=5`

#### APPEARS IN

#### RELATED QUESTIONS

Two circles touch each other externally at P. AB is a common tangent to the circles touching them at A and B. The value of ∠ L APB is

(A) 30°

(B) 45°

(C) 60°

(D) 90°

In Fig. 1, QR is a common tangent to the given circles, touching externally at the point T. The tangent at T meets QR at P. If PT = 3.8 cm, then the length of QR (in cm) is :

(A) 3.8

(B) 7.6

(C) 5.7

(D) 1.9

n Fig. 2, PQ and PR are two tangents to a circle with centre O. If ∠QPR = 46°, then ∠QOR equals:

(A) 67°

(B) 134°

(C) 44°

(D) 46°

Prove that the line segment joining the point of contact of two parallel tangents to a circle is a diameter of the circle.

In the given figure, the incircle of ∆ABC touches the sides BC, CA and AB at D, E, F respectively. Prove that AF + BD + CE = AE + CD + BF = `\frac { 1 }{ 2 } ("perimeter of ∆ABC")`

Prove that there is one and only one tangent at any point on the circumference of a circle.

In Fig., if AB = AC, prove that BE = EC

PA and PB are tangents from P to the circle with centre O. At point M, a tangent is drawn cutting PA at K and PB at N. Prove that KN = AK + BN.

Write True or False. Give reasons for your answers.

If a circle is divided into three equal arcs, each is a major arc.

Write True or False. Give reason for your answer.

Sector is the region between the chord and its corresponding arc.

In figure OQ : PQ = 3 : 4 and perimeter of ΔPDQ = 60cm. determine PQ, QR and OP.

true or false

A circle is a plane figure.

Draw different pairs of circles. How many points does each pair have in common? What is the maximum number of common points?

In the following figure, OABC is a square. A circle is drawn with O as centre which meets OC

at P and OA at Q. Prove that:

(i) ΔOPA ≅ ΔOQC, (ii) ΔBPC ≅ ΔBQA.

In the given figure, if arc AB = arc CD, then prove that the quadrilateral ABCD is an isosceles– trapezium (O is the centre of the circle).

A circle is inscribed in a ΔABC touching AB, BC and AC at P, Q and R respectively. If AB = 10 cm, AR=7cm and CR=5cm, find the length of BC.

In the given figure, O is the centre of the two concentric circles of radii 4 cm and 6cm respectively. AP and PB are tangents to the outer and inner circle respectively. If PA = 10cm, find the length of PB up to one place of the decimal.

In the given figure, two tangents RQ, and RP and RP are drawn from an external point R to the circle with centre O. If ∠PRQ =120° , then prove that OR = PR + RQ.

In Fig. 4, an isosceles triangle ABC, with AB = AC, circumscribes a circle. Prove that the point of contact P bisects the base BC.

A quadrilateral ABCD is drawn to circumscribe a circle. Prove that AB + CD = AD + BC ?

The circumference of a circle is 22 cm. The area of its quadrant (in cm^{2}) is

In a cyclic quadrilateral *ABCD* if *AB* || *CD* and ∠*B** *= 70°, find the remaining angles.

If the length of a chord of a circle is 16 cm and is at a distance of 15 cm from the centre of the circle, then the radius of the circle is

The radius of a circle is 6 cm. The perpendicular distance from the centre of the circle to the chord which is 8 cm in length, is

Number of circles that can be drawn through three non-collinear points is

If \[d_1 , d_2 ( d_2 > d_1 )\] be the diameters of two concentric circle s and *c* be the length of a chord of a circle which is tangent to the other circle , prove that\[{d_2}^2 = c^2 + {d_1}^2\].

Choose correct alternative answer and fill in the blank.

Radius of a circle is 10 cm and distance of a chord from the centre is 6 cm. Hence the length of the chord is .........

Radius of a circle is 10 cm and distance of a chord from the centre is 6 cm. Hence the length of the chord is ______.

The point of concurrence of all angle bisectors of a triangle is called the ______.

The circle which passes through all the vertices of a triangle is called ______.

Length of a chord of a circle is 24 cm. If distance of the chord from the centre is 5 cm, then the radius of that circle is ______.

The length of the longest chord of the circle with radius 2.9 cm is ______.

Radius of a circle with centre O is 4 cm. If l(OP) = 4.2 cm, say where point P will lie.

The lengths of parallel chords which are on opposite sides of the centre of a circle are 6 cm and 8 cm. If radius of the circle is 5 cm, then the distance between these chords is ______.

Find the length of the chord of a circle in the following when:

Radius is 13 cm and the distance from the centre is 12 cm

Find the area of a circle of radius 7 cm.

In the given figure, O is the centre of a circle, chord PQ ≅ chord RS If ∠ POR = 70° and (arc RS) = 80°, find –

(1) m(arc PR)

(2) m(arc QS)

(3) m(arc QSR)

In the given figure, chord EF || chord GH. Prove that, chord EG ≅ chord FH. Fill in the blanks and write the proof.

**The figure given below shows a circle with center O in which diameter AB bisects the chord CD at point E. If CE = ED = 8 cm and EB = 4 cm,**

find the radius of the circle.

**In the following figure, OABC is a square. A circle is drawn with O as centre which meets OC at P and OA at Q.**

Prove that:

( i ) ΔOPA ≅ ΔOQC

( ii ) ΔBPC ≅ ΔBQA

**Draw two circles of different radii. How many points these circles can have in common? What is the maximum number of common points?**

**Suppose you are given a circle. Describe a method by which you can find the center of this circle.**

In the above figure, seg AB is a diameter of a circle with centre P. C is any point on the circle. seg CE ⊥ seg AB. Prove that CE is the geometric mean of AE and EB. Write the proof with the help of the following steps:

a. Draw ray CE. It intersects the circle at D.

b. Show that CE = ED.

c. Write the result using the theorem of the intersection of chords inside a circle. d. Using CE = ED, complete the proof.

In an equilateral triangle, prove that the centroid and center of the circum-circle (circumcentre) coincide.

In the given circle with diameter AB, find the value of x.

In Fig., chords AB and CD of the circle intersect at O. AO = 5 cm, BO = 3 cm and CO = 2.5 cm. Determine the length of DO.

If O is the centre of the circle, find the value of x in each of the following figures

If O is the centre of the circle, find the value of x in each of the following figures

ABC is a triangle with AB = 10 cm, BC = 8 cm and AC = 6 cm (not drawn to scale). Three circles are drawn touching each other with the vertices as their centres. Find the radii of the three circles.

**Use the figure given below to fill in the blank:**

Tangent to a circle is _______.

**Use the figure given below to fill in the blank:**

If the length of RS is 5 cm, the length of PQ = _______

**Use the figure given below to fill in the blank:**

AB is a ______ of the circle.

Draw circle with diameter: 6 cm

In above case, measure the length of the radius of the circle drawn.

Draw a circle of radius 6 cm. In the circle, draw a chord AB = 6 cm.

(i) If O is the center of the circle, join OA and OB.

(ii) Assign a special name to ∆AOB

(iii) Write the measure of angle AOB.

Draw a circle of radius 3.6 cm. In the circle, draw a chord AB = 5 cm. Now shade the minor segment of the circle.

Construct a triangle PQR with QR = 5.5 cm, ∠Q = 60° and angle R = 45°. Construct the circumcircle cif the triangle PQR.

The diameter of a circle is 12.6 cm. State, the length of its radius.

**State, if the following statement is true or false:**

The longest chord of a circle is its diameter.

**State, if the following statement is true or false:**

Every diameter bisects a circle and each part of the circle so obtained is a semi-circle.

If the radius of a circle is 5 cm, what will its diameter be?

**Draw circle with the radii given below.**

2 cm

**Draw circle with the radii given below.**

3 cm

**Draw a circle with the radii given below.**

4 cm

Draw a circle of any radius. Show one diameter, one radius, and one chord on that circle.

In the table below, write the names of the points in the interior and exterior of the circle and those on the circle.

Diagram |
Points in the interior of the circle |
Points in the exterior of the circle |
Points on the circle |

The diameter of the circle is 52 cm and the length of one of its chord is 20 cm. Find the distance of the chord from the centre

The chord of length 30 cm is drawn at the distance of 8 cm from the centre of the circle. Find the radius of the circle

Find the length of the chord AC where AB and CD are the two diameters perpendicular to each other of a circle with radius `4sqrt(2)` cm and also find ∠OAC and ∠OCA

A chord is 12 cm away from the centre of the circle of radius 15 cm. Find the length of the chord

In a circle, AB and CD are two parallel chords with centre O and radius 10 cm such that AB = 16 cm and CD = 12 cm determine the distance between the two chords?

Two circles of radii 5 cm and 3 cm intersect at two points and the distance between their centres is 4 cm. Find the length of the common chord

A chord is at a distance of 15 cm from the centre of the circle of radius 25 cm. The length of the chord is

In the figure, O is the centre of a circle and diameter AB bisects the chord CD at a point E such that CE = ED = 8 cm and EB = 4 cm. The radius of the circle is

AD is a diameter of a circle and AB is a chord If AD = 30 cm and AB = 24 cm then the distance of AB from the centre of the circle is

The ratio between the circumference and diameter of any circle is _______

A line segment which joins any two points on a circle is a ___________

The longest chord of a circle is __________

The radius of a circle of diameter 24 cm is _______

A part of circumference of a circle is called as _______

Find the missing values in the following table for the circles with radius (r), diameter (d) and Circumference (C).

radius (r) |
diameter (d) |
Circumference (C) |

15 cm |

Find the missing values in the following table for the circles with radius (r), diameter (d) and Circumference (C).

radius (r) |
diameter (d) |
Circumference (C) |

1760 cm |

Find the missing values in the following table for the circles with radius (r), diameter (d) and Circumference (C).

radius (r) |
diameter (d) |
Circumference (C) |

24 m |

All the radii of a circle are _______________

The ______________ is the longest chord of a circle

A line segment joining any point on the circle to its center is called the _____________ of the circle

A line segment with its end points on the circle is called a ______________

Twice the radius is ________________

Find the diameter of the circle

Radius = 10 cm

Find the diameter of the circle

Radius = 8 cm

Find the radius of the circle

Diameter = 24 cm

Find the radius of the circle

Diameter = 30 cm

Find the radius of the circle

Diameter = 76 cm

A, B, C are any points on the circle with centre O. If m(arc BC) = 110° and m(arc AB) = 125°, find measure arc AC.

In the adjoining figure, seg DE is the chord of the circle with center C. seg CF⊥ seg DE and DE = 16 cm, then find the length of DF?

In figure, O is the centre of a circle, chord PQ ≅ chord RS. If ∠POR = 70° and (arc RS) = 80°, find

(i) m(arc PR)

(ii) m(arc QS)

(iii) m(arc QSR)

In the figure, segment PQ is the diameter of the circle with center O. The tangent to the tangent circle drawn from point C on it, intersects the tangents drawn from points P and Q at points A and B respectively, prove that ∠AOB = 90°

**Given:** A circle inscribed in a right angled ΔABC. If ∠ACB = 90° and the radius of the circle is r.

**To prove:** 2r = a + b – c

In a circle with centre P, chord AB is parallel to a tangent and intersects the radius drawn from the point of contact to its midpoint. If AB = `16sqrt(3)`, then find the radius of the circle

In the figure, O is the center of the circle. Line AQ is a tangent. If OP = 3, m(arc PM) = 120°, then find the length of AP.

In the figure, O is the centre of the circle, and ∠AOB = 90°, ∠ABC = 30°. Then find ∠CAB.

In the figure, a circle touches all the sides of quadrilateral ABCD from the inside. The center of the circle is O. If AD⊥ DC and BC = 38, QB = 27, DC = 25, then find the radius of the circle.

Circles with centres A, B and C touch each other externally. If AB = 36, BC = 32, CA = 30, then find the radii of each circle.

In the following figure, tangents PQ and PR are drawn to a circle such that ∠RPQ = 30°. A chord RS is drawn parallel to the tangent PQ, then ∠RQS.

Three circles touch each other externally. The distance between their centres is 5 cm, 6 cm, and 7 cm. Find the radii of the circles.

In the given figure, AB is the diameter of the circle. Find the value of ∠ACD.

If AB = 12 cm, BC = 16 cm and AB is perpendicular to BC, then the radius of the circle passing through the points A, B and C is ______.

On a common hypotenuse AB, two right triangles ACB and ADB are situated on opposite sides. Prove that ∠BAC = ∠BDC.

Draw two acute angles and one obtuse angle without using a protractor. Estimate the measures of the angles. Measure them with the help of a protractor and see how much accurate is your estimate

In the given figure, O is the centre of the circle. Name all chords of the circle.

From the figure, identify three radii.

From the figure, identify a diameter.

From the figure, identify two points in the interior.

From the figure, identify a point in the exterior.

From the figure, identify a segment.

Is every diameter of a circle also a chord?

Is every chord of a circle also a diameter?

Say true or false:

The centre of a circle is always in its interior.

A figure is in the form of rectangle PQRS having a semi-circle on side QR as shown in the figure. Determine the area of the plot.

A circle of radius 3 cm with centre O and a point L outside the circle is drawn, such that OL = 7 cm. From the point L, construct a pair of tangents to the circle. Justify LM and LN are the two tangents.

A 7 m broad pathway goes around a circular park with a circumference of 352 m. Find the area of road.

The circumcentre of a triangle is the point which is ______.