Secant

Q.

Two tangents PA and PB are drawn to a circle with center O from an external point P. Prove that $\angle APB=2\angle OAB$ [3 MARKS]

Q.

A line which cuts a circle at two distinct points is called the __ of a circle.

Q. In a circle, two chords AB and CD intersect at a point inside the circle. Prove that
$$\begin{gathered} \left(\mathrm{a}\right)△\mathrm{PAC}~△\mathrm{PDB} \\ \left(\mathrm{b}\right)\mathrm{PA}.\mathrm{PB}=\mathrm{PC}.\mathrm{PD} \end{gathered}$$

Q.

In the given figure, PT is a common tangent to the circles touching externally at P and AB is another common tangent touching the circles at A and B. Prove that: [3 MARKS]

(i) T is the mid-point of AB

$\left(ii\right)\angle APB={90}^{\circ }$

(iii) If X and Y are centers of the two circles, show that the circle on AB as diameter touches the line XY.

Q. Prove that the tangent at any point of a circle is perpendicular to the radius through the point of contact

Q.

The angle between the curves $y=a^x$ and $y=b^x$ is equal to

1. ${\tan }^{-1}\left|\frac{\left(a-b\right)}{1+ab}\right|$

2. ${\tan }^{-1}\left|\frac{\left(a-b\right)}{1-ab}\right|$

3. ${\tan }^{-1}\left|\frac{\left(\log b+\log a\right)}{1+\log a\log b}\right|$

4. ${\tan }^{-1}\left|\frac{\left(\log b+\log a\right)}{1-\log a\log b}\right|$

5. ${\tan }^{-1}\left|\frac{\left(\log b-\log a\right)}{1+\log a\log b}\right|$

Q. If two circles are touching externally, how many common tangents of them can be drawn?

1. One
2. Two
3. Three
4. Four

Q.

Two chords AB and AC of a circle are equal. Prove that the centre of the circle lies on the bisector of angle BAC.

Show that the bisector of angle BAC is a perpendicular bisector of chord BC

Q. 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)

Q. Question 7
In figure, tangents PQ and PR are drawn to a circle such that $\angle RPQ={30}^{\circ }$. A chord RS is drawn parallel to the tangent PQ. Find the $\angle$ RQS.

Q.

In the figure, PT touches a circle with center O at R. Diameter SQ when produced meets RT at P. ∠SPR = ^o, ∠QRP = y^o, find the value of ^o + 2y^o.

1. 180^o

2. 360^o

3. 270^o

4. 90^o

5. 30^o

Q. Question 10
AB is a diameter of a circle and AC is the chord such that $\angle$ BAC = ${30}^{\circ }$. If the tangent at C intersects AB extended at D, then BC = BD.

Q. ______ is a line that touches a circle at only one point, and ______ is a line that intersects a circle at two distinct points.

1. Tangent, secant
2. Secant, secant
3. Secant, tangent
4. Tangent, tangent

Q. ______ is a line that touches a circle at only one point, and ______ is a line that intersects a circle at two distinct points.

1. Secant, secant
2. Secant, tangent
3. Tangent, tangent
4. Tangent, secant

Q.

In the given figure, two chords AB and CD intersect each other at the point P. prove that:

(i) ΔAPC ∼ ΔDPB

(ii) AP.BP = CP.DP

Q.

Which of the following statements is/are correct?

1. With respect to a tangent both the circles lie on the same side, this tangent is called direct common tangent.

2. With respect to a tangent both the circles lie on the opposite side, this tangent is called transverse (indirect) common tangent.

1. Only 2

2. Only 1

3. Both 1 & 2

4. None of these

Q. Question 5
At one end A of a diameter AB of a circle of radius 5 cm, tangent XAY is drawn to the circle. The length of the chord CD parallel to XY and at a distance 8 cm from A, is
(A) 4 cm
(B) 5 cm
(C) 6 cm
(D) 8 cm

Q.

A line which cuts a circle at two distinct points is called the __ of a circle.

Q.

A line that intersects a circle in two distinct points is called a

1. Tangent

2. Diameter

3. Chord

4. Secant

Q. In figure. If PQR is the tangent to a circle at Q whose centre is O, AB is a chord parallel to PR and $\angle BQR={70}^{\circ }$ , then $\angle$AQB is equal to

Q.

The angle of elevation of the top of a tower from a point P on the ground is α. After walking a distance d towards the foot of the tower, angle of elevation is found to be β. Then:

1. None of these

2. α > β

3. α = β

4. α < β

Q. Explain the theorem of tangent line

Q. is a line that touches a circle at only one point

1. Tangent
2. Secant
3. Chord

Q. is a line that touches a circle at only one point

1. Tangent
2. Secant
3. Chord

Q.

When a chord divides a circle into two parts, each part is called a:

1. Chord

2. Segment

3. Secant

4. Centre

Q. A line that touches a circle at exactly one point is called a ______.

1. tangent
2. secant
3. diameter
4. radius

Q.

___ is a line that intersects a circle at only one point, and ___ is a line that intersects a circle at two distinct points.

1. Tangent, secant

2. Secant, secant

3. Secant, tangent

4. Tangent tangent

Q.

A___ is a line and a ___ is a line segment.

1. secant, chord

2. chord , secant

3. diameter, radius

4. radius, diameter

Q. ______ is a line that touches a circle at only one point, and ______ is a line that intersects a circle at two distinct points.

1. Tangent, secant
2. Secant, secant
3. Secant, tangent
4. Tangent, tangent

Q. Prove that “ The perpendicular from the centre of a circle to a chord bisects the chord”