Mathematics · Circles · NCERT Exercises

Exercise 10.1

Complete, independently verified solutions for NCERT Exercise 10.1.

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Mathematics · Chapter 10 · NCERT Exercises

Verified NCERT questions and solutionsSolutions are checked independently and follow the supplied NCERT chapter edition.

Question

Question 1

number of tangents

How many tangents can a circle have?

Solution

Board-exam working:

A tangent can be drawn at every point of a circle.

A circle contains infinitely many points.

\therefore\ \text{a circle has infinitely many tangents}

Final answer

Infinitely many tangents.

Question

Question 2

circle terminology · fill in the blanks

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 at most ___ parallel tangents. (iv) The common point of a tangent and the circle is called the ___.

Solution

Board-exam working:

(i) A tangent touches the circle at exactly one point.

\text{Answer: one}

(ii) A line meeting a circle at two points is a secant.

\text{Answer: secant}

(iii) Two tangents can be parallel, one on each side of the circle.

\text{Answer: two}

(iv) Their common point is the point of contact.

\text{Answer: point of contact}

Final answer

(i) one; (ii) secant; (iii) two; (iv) point of contact.

Question

Question 3

Pythagoras theorem · tangent length

A tangent PQ at point P of a circle of radius 5 cm meets a line through the centre O at Q, where OQ = 12 cm. Find PQ. Options: (A) 12 cm, (B) 13 cm, (C) 8.5 cm, (D) √119 cm.

Solution

Board-exam working:

Circle with centre O; P is the point of contact, Q is external, and the radius is perpendicular to the tangent. — Radius and tangent form a right triangle — solution sketch (not to scale).

The radius to the point of contact is perpendicular to the tangent.

OP\perp PQ

In right triangle OPQ:

OQ^2=OP^2+PQ^2

12^2=5^2+PQ^2

PQ^2=144-25=119

PQ=\sqrt{119}\text{ cm}

Final answer

Option (D): √119 cm.

Question

Question 4

construction · tangent and secant

Draw a circle and two lines parallel to a given line such that one is a tangent and the other is a secant to the circle.

Solution

Board-exam working:

The given line ℓ, tangent ℓ₁ and secant ℓ₂ are horizontal and parallel. ℓ₁ touches at P; ℓ₂ cuts the circle twice. — Parallel tangent and secant — solution sketch (not to scale).

Draw a circle with centre O and radius r. Let ℓ be the given line.

Draw a line ℓ₁ parallel to ℓ at perpendicular distance r from O.

d(O,\ell_1)=r

Since ℓ₁ meets the circle at exactly one point, ℓ₁ is a tangent.

Draw another line ℓ₂ parallel to ℓ with its perpendicular distance from O less than r.

d(O,\ell_2)<r

The line ℓ₂ cuts the circle at two points, so ℓ₂ is a secant.

\ell_1\parallel\ell_2\parallel\ell

Final answer

The required parallel lines are ℓ₁, the tangent, and ℓ₂, the secant.

Exercise 10.1

Choose a question

Question 1

Solution

Final answer

Question 2

Solution

Final answer

Question 3

Solution

Final answer

Question 4

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