Tangent Circles

Q.

If a triangle $ABC$has vertices $A(-1,7),B(-7,1)$, and$C(5,-5)$, then its orthocentre has coordinates

1. $\left(-3,3\right)$

2. $\left(-3/5,3/5\right)$

3. $\left(3/5,-3/5\right)$

4. $(3,-3)$

Q.

The number of tangents that can be drawn from a point outside a circle is ____.

1. 1

2. 2

3. 0

4. 3

Q.

How many tangents can be drawn from a point which lies on a circle ?

1. Q

2. Both Q and R

3. Not possible through both Q and R

4. R

Q. If four sides of a quadrilateral ABCD are tangential to a circle, then

(a) AC + AD = BD + CD

(b) AB + CD = BC + AD

(c) AB + CD = AC + BC

(d) AC + AD = BC + DB

Q.

Two circles of radii 5 cm and 3 cm are concentric. Calculate the length of a chord of the outer circle which touches the inner circle.

1. 12 cm

2. 23 cm

3. 2 cm

4. 8 cm

Q.

The circle ${\mathrm{x}}^2+{\mathrm{y}}^2=4\mathrm{x}+8\mathrm{y}+5$ intersects the line $3\mathrm{x}-4\mathrm{y}=\mathrm{m}$ at two distinct points, if

1. $-85<\mathrm{m}<–35$

2. $-35<\mathrm{m}<15$

3. $15<\mathrm{m}<65$

4. $35<\mathrm{m}<85$

Q. In the given figure, AB and AC are tangents to a circle with centre O and radius 8 cm. If OA = 17 cm, then the length of AC (in cm) is
(a) 9 cm
(b) 15 cm
(c) $\sqrt{353}$ cm
(d) 25 cm

Q.

The equation of the pair of straight lines joining the point of intersection of the curve $x^2+y^2=4$ and $y-x=2$ to the origin, is:

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

2. $x^2+y^2+{(y-x)}^2=0$

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

4. $x^2+y^2+4{(y-x)}^2=0$

Q.

In a $∆ABC,\text{if}a=3,b=4,c=5$, then the distance between its incenter and circumcenter is

1. $\frac{1}{2}$

2. $\frac{\sqrt{3}}{2}$

3. $\frac{3}{2}$

4. $\frac{\sqrt{5}}{2}$

Q. A quadrilateral ABCD is drawn to circumscribe a circle Such that its sides AB, Bc, CD and AD touchs the circle at P, Q, R , and S respectively. If AB=xcm , BC=7cm, CR=3cm and As=5cm, find x

Q.

A doorway is decorated as shown in the figure. There are 4 semi circles. BC, the diameter of the larger circle is of length 84 cm. Centres of 3 semi circles lie on BC. ∆ ABC is isosceles with AB = AC. If BO = OC, find the area of shaded region.

1. 1092 cm²

2. 1950 cm²

3. 1100 cm²

4. 1932 cm²

5. 2000 cm²

Q.

In the figure PA and PB are tangents, . What is the measure of ?

Q.

A triangle is formed by the lines whose combined equation is given by $\left(x+y-4\right)\left(xy-x-y+2\right)=0$ The equation of its circumcircle is

1. $x^2+y^2–5x–3y+8=0$

2. $x^2+y^2–3x–5y+8=0$

3. $x^2+y^2–3x–5y-8=0$

4. none of these

Q. Question 8
A quadrilateral ABCD is drawn to circumscribe a circle. Prove that AB + CD = AD + BC.

Q.

The value of $\frac{1}{{r_1}^2}+\frac{1}{{r_2}^2}+\frac{1}{{r_3}^2}+\frac{1}{r^2}$ is

1. $\frac{\left(a^2+b^2+c^2\right)}{{∆}^2}$

2. $0$

3. $1$

4. $\frac{{∆}^2}{\left(a^2+b^2+c^2\right)}$

Q.

The tangent makes a _____ angle with the radius at the point of contact.

1. obtuse

2. right

3. reflex

4. acute

Q.

If tangents PA and PB from a point P to a circle with centre O are inclined to each other an angle of 80, then ∠POA is equal to

(A) 50 (B) 60

(C) 70 (D) 80

Q.

Question 3
State whether the following is true or false.
A pair of tangents can be constructed from a point P to a circle of radius 3.5 cm situated at a distance of 3 cm from the centre.
[Thinking process
Let r = radius of circle and d = distance of a point from the centre.
1. If r = d, then point lie on the circle i.e only one tangent is possible
2. If r < d, then point lie outside the circle. i.e a pair of tangent is possible.
3. If r>d, then point lie inside the circle i.e no tangent is possible.
]

Q. If the sides of a quadrilateral touch a circle. prove that the sum of a pair of opposite sides is equal to the sum of the other pair.

Q. In the given figure, AB and AC are tangents to a circle with centre O such that ∠BAC = 40^∘ .Then ∠BOC is equal to [CBSE 2011, 14]
(a) 80^∘
(b) 100^∘
(c) 120^∘
(d) 140^∘

Q.

For the given quadrilateral, find BC + AD.

1. 20 cm

2. 15 cm

3. 19 cm

4. 14 cm

Q.

The centres of two circles of radii 3 cm and 2 cm are 8 cm apart. Find the length of the common tangent.

1. $\sqrt{39}cm$

2. 64 cm

3. 39 cm

4. 25 cm

Q.

In the given figure, quadrilateral ABCD circumscribes a circle, as shown. Then, AB + CD = ___.

1. 2( AD + BC)

2. BC

3. AD

4. AD + BC

Q. AB is a diameter and AC is a chord of a circle with centre O such that angle BAC = 30°. The tangent at C intersects extended AB at a point D. Prove that BC=BD.

Q.

In the figure. What measure of the makes AP a tangent?

Q. How many tangents can be drawn to a circle from a point on the same circle? Justify your answer.

Q. How many tangents can be drawn from
(i) a point which lies on a circle
(ii) a point which lies outside the circle
[1 MARK]

Q.

Prove that opposite sides of a quadrilateral circumscribing a circle subtend supplementary angles at the centre of the circle.

Q.

A tangent PQ at a point P of a circle of radius 5 cm meets a line through the centre O at a point Q so that OQ = 12 cm. Length PQ is :

(A) 12 cm. (B) 13 cm (C) 8.5 cm (D) cm test

Q.

How many tangents can be drawn to a circle through the points A, B and C as shown?

1. 0, 1, 2

2. 2, 1, 0

3. 1, 0, 2

4. 2, 0, 1