EXERCISE 10.1
Circles • 4 Questions
Question 1
Hint available
How many tangents can a circle have?
Key Idea
A tangent to a circle is a straight line that touches the circle at exactly one point. Since every point on the circumference can serve as a point of contact, and there are infinitely many points on the circumference, the circle can have infinitely many distinct tangents.
Step-by-Step Solution
1. Definition of a tangent: A line is called a tangent to a circle if it meets the circle at exactly one point (the point of contact).\
2. Point of contact: Any point on the circumference of the circle can be chosen as a point of contact.\
3. Number of points on the circumference: The set of points on a circle is infinite (uncountably many).\
4. Correspondence: For each distinct point of contact there exists a unique tangent line (the line perpendicular to the radius drawn to that point).\
5. Conclusion: Hence, as there are infinitely many points on the circle, there are infinitely many distinct tangents that can be drawn to the circle.
Therefore, a circle can have infinitely many tangents.
2. Point of contact: Any point on the circumference of the circle can be chosen as a point of contact.\
3. Number of points on the circumference: The set of points on a circle is infinite (uncountably many).\
4. Correspondence: For each distinct point of contact there exists a unique tangent line (the line perpendicular to the radius drawn to that point).\
5. Conclusion: Hence, as there are infinitely many points on the circle, there are infinitely many distinct tangents that can be drawn to the circle.
Therefore, a circle can have infinitely many tangents.
Question 2
Hint available
Fill in the blanks : (i) A tangent to a circle intersects it in point (s). (ii) A line intersecting a circle in two points is called a . (iii) A circle can have parallel tangents at the most. (iv) The common point of a tangent to a circle and the circle is called .
Key Idea
Recall the basic definitions related to a circle: • Tangent – a straight line that touches the circle at exactly one point. The point of touching is called the *point of contact* (or point of tangency). • Secant – a straight line that cuts the circle at two distinct points. • At most two tangents to a given circle can be parallel to each other (e.g., the top and bottom horizontal tangents).
Step-by-Step Solution
1. Tangent and its point of contact
- By definition, a tangent touches the circle at exactly one point. That unique point is termed the *point of contact* (also called point of tangency). Hence, the blank (i) is filled with point of contact.
2. Line intersecting a circle at two points
- A straight line that meets the circle in two distinct points is called a *secant*. Therefore, blank (ii) is secant.
3. Maximum number of parallel tangents
- Through any circle, at most two tangents can be drawn that are parallel to each other (think of the horizontal tangents at the top and bottom of the circle). Hence, blank (iii) is two.
4. Common point of a tangent and the circle
- The single point where a tangent touches the circle is again the *point of contact*. Thus, blank (iv) is point of contact.
Putting all the answers together:
(i) point of contact
(ii) secant
(iii) two
(iv) point of contact
- By definition, a tangent touches the circle at exactly one point. That unique point is termed the *point of contact* (also called point of tangency). Hence, the blank (i) is filled with point of contact.
2. Line intersecting a circle at two points
- A straight line that meets the circle in two distinct points is called a *secant*. Therefore, blank (ii) is secant.
3. Maximum number of parallel tangents
- Through any circle, at most two tangents can be drawn that are parallel to each other (think of the horizontal tangents at the top and bottom of the circle). Hence, blank (iii) is two.
4. Common point of a tangent and the circle
- The single point where a tangent touches the circle is again the *point of contact*. Thus, blank (iv) is point of contact.
Putting all the answers together:
(i) point of contact
(ii) secant
(iii) two
(iv) point of contact
Question 3
Hint available
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) 119 cm.
Key Idea
The radius drawn to the point of tangency is perpendicular to the tangent. Hence, triangle OPQ is a right‑angled triangle with right angle at P. Use the Pythagorean theorem to relate the sides OP (radius), OQ (given) and PQ (required).
Step-by-Step Solution
1. Draw the circle with centre O and radius OP = 5 cm.
2. At point P draw the tangent PQ.
3. Join O to Q; the line OQ passes through the centre and meets the tangent at Q.
4. Since a radius to a point of tangency is perpendicular to the tangent, \(OP \perp PQ\).
5. Therefore, triangle \(OPQ\) is right‑angled at \(P\).
6. Apply the Pythagorean theorem:
$$OP^{2} + PQ^{2} = OQ^{2}$$
Substituting the known lengths:
$$5^{2} + PQ^{2} = 12^{2}$$
$$25 + PQ^{2} = 144$$
$$PQ^{2} = 144 - 25 = 119$$
7. Hence,
$$PQ = \sqrt{119}\ \text{cm}$$
8. The numerical value of \(\sqrt{119}\) is approximately 10.9 cm, which is not listed among the options; the closest representation is option (D) "119 cm" interpreted as \(\sqrt{119}\) cm.
2. At point P draw the tangent PQ.
3. Join O to Q; the line OQ passes through the centre and meets the tangent at Q.
4. Since a radius to a point of tangency is perpendicular to the tangent, \(OP \perp PQ\).
5. Therefore, triangle \(OPQ\) is right‑angled at \(P\).
6. Apply the Pythagorean theorem:
$$OP^{2} + PQ^{2} = OQ^{2}$$
Substituting the known lengths:
$$5^{2} + PQ^{2} = 12^{2}$$
$$25 + PQ^{2} = 144$$
$$PQ^{2} = 144 - 25 = 119$$
7. Hence,
$$PQ = \sqrt{119}\ \text{cm}$$
8. The numerical value of \(\sqrt{119}\) is approximately 10.9 cm, which is not listed among the options; the closest representation is option (D) "119 cm" interpreted as \(\sqrt{119}\) cm.
Question 4
Hint available
Draw a circle and two lines parallel to a given line such that one is a tangent and the other, a secant to the circle.
Key Idea
Use the definition of a tangent (a line touching a circle at exactly one point) and a secant (a line intersecting a circle at two distinct points). By constructing a line parallel to a given line, we can adjust its position so that it meets the circle either as a tangent or as a secant.
Step-by-Step Solution
1. Draw the given line: With a ruler, draw a straight line $l$ on the paper. This line will serve as the reference for parallelism.
2. Choose a point for the centre: Select any point $O$ not on line $l$ and mark it as the centre of the required circle.
3. Draw the circle: Using a compass, place the needle at $O$, choose a convenient radius $r$, and draw the circle $C$.
4. Construct a tangent parallel to $l:
- From the centre $O$, draw a perpendicular to the given line $l$; let the foot of the perpendicular be $H$.
- Measure the distance $OH$.
- On the same side of $l$ as the circle, mark a point $P$ on the line through $H$ such that $HP = OH$ (i.e., translate the perpendicular distance to the other side of $l$). The line through $P$ parallel to $l$ will be at a distance $r$ from $O$.
- With a ruler, draw the line $t$ through $P$ parallel to $l$. Since the distance from $O$ to $t$ equals the radius $r$, $t$ touches the circle at exactly one point – it is the required tangent.
5. Construct a secant parallel to $l:
- Choose a point $Q$ on the same side of $l$ such that the perpendicular distance from $O$ to the line through $Q$ parallel to $l$ is less than $r$ (for example, take $Q$ on the line through $H$ but nearer to $l$ than $P$).
- Draw the line $s$ through $Q$ parallel to $l$. Because the distance from $O$ to $s$ is less than the radius, the line $s$ cuts the circle at two points $A$ and $B$ – hence $s$ is a secant.
6. Verification:
- Check that $t$ touches the circle at only one point (use a compass to confirm no second intersection).
- Verify that $s$ intersects the circle at two distinct points.
7. Label the figure: Mark the centre $O$, the tangent $t$, the secant $s$, the points of contact $T$ (for tangent) and $A, B$ (for secant), and indicate that $t \parallel s \parallel l$.
The construction satisfies the requirement: two lines parallel to the given line, one tangent and the other a secant to the drawn circle.
2. Choose a point for the centre: Select any point $O$ not on line $l$ and mark it as the centre of the required circle.
3. Draw the circle: Using a compass, place the needle at $O$, choose a convenient radius $r$, and draw the circle $C$.
4. Construct a tangent parallel to $l:
- From the centre $O$, draw a perpendicular to the given line $l$; let the foot of the perpendicular be $H$.
- Measure the distance $OH$.
- On the same side of $l$ as the circle, mark a point $P$ on the line through $H$ such that $HP = OH$ (i.e., translate the perpendicular distance to the other side of $l$). The line through $P$ parallel to $l$ will be at a distance $r$ from $O$.
- With a ruler, draw the line $t$ through $P$ parallel to $l$. Since the distance from $O$ to $t$ equals the radius $r$, $t$ touches the circle at exactly one point – it is the required tangent.
5. Construct a secant parallel to $l:
- Choose a point $Q$ on the same side of $l$ such that the perpendicular distance from $O$ to the line through $Q$ parallel to $l$ is less than $r$ (for example, take $Q$ on the line through $H$ but nearer to $l$ than $P$).
- Draw the line $s$ through $Q$ parallel to $l$. Because the distance from $O$ to $s$ is less than the radius, the line $s$ cuts the circle at two points $A$ and $B$ – hence $s$ is a secant.
6. Verification:
- Check that $t$ touches the circle at only one point (use a compass to confirm no second intersection).
- Verify that $s$ intersects the circle at two distinct points.
7. Label the figure: Mark the centre $O$, the tangent $t$, the secant $s$, the points of contact $T$ (for tangent) and $A, B$ (for secant), and indicate that $t \parallel s \parallel l$.
The construction satisfies the requirement: two lines parallel to the given line, one tangent and the other a secant to the drawn circle.