✦ NCF 2023 ALIGNED • CBSE CLASS X ✦
Chapter 10: Circles
Assertion-Reason Question Bank
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Assertion (A)
A line intersecting a circle in two points is called a tangent.
Reason (R)
A line intersecting a circle in two distinct points is called a secant.
Correct Option: (D)
Assertion (A) is false. Reason (R) is true.
Assertion (A) is false. Reason (R) is true.
Correct Answer: (D)
Assertion false. Reason (R) is true.
Assertion false. Reason (R) is true.
Assertion (A)
Distance between two parallel tangents of radius \(r\) circle is \(r\).
Reason (R)
Distance between parallel tangents is equal to diameter \(2r\).
Correct Option: (D)
Assertion (A) is false (it is \(2r\), not \(r\)). Reason (R) is true.
Assertion (A) is false (it is \(2r\), not \(r\)). Reason (R) is true.
Correct Answer: (D)
Assertion false. Reason (R) is true.
Assertion false. Reason (R) is true.
Assertion (A)
If angle between two tangents is \(90^\circ\) and radius is \(5\text{ cm}\), distance of external point from centre is \(5\sqrt{2}\text{ cm}\).
Reason (R)
In quadrilateral \(OAPB\), \(OAPB\) forms a rectangle with unequal adjacent sides.
Correct Option: (C)
Assertion (A) is true (\(OP = r\sqrt{2} = 5\sqrt{2}\text{ cm}\)). Reason (R) is false because \(OAPB\) forms a square with all sides \(5\text{ cm}\).
Assertion (A) is true (\(OP = r\sqrt{2} = 5\sqrt{2}\text{ cm}\)). Reason (R) is false because \(OAPB\) forms a square with all sides \(5\text{ cm}\).
Correct Answer: (C)
Assertion (A) is true. Reason (R) is false.
Assertion (A) is true. Reason (R) is false.
Assertion (A)
Tangent to a circle can intersect the circle at two points.
Reason (R)
A tangent touches the circle at exactly one point called the point of contact.
Correct Option: (D)
Assertion (A) is false. Reason (R) is true.
Assertion (A) is false. Reason (R) is true.
Correct Answer: (D)
Assertion false. Reason (R) is true.
Assertion false. Reason (R) is true.
Assertion (A)
Number of tangents from a point inside a circle is \(0\).
Reason (R)
Number of tangents from a point on the circle is \(2\).
Correct Option: (C)
Assertion (A) is true (no tangent can be drawn from inside). Reason (R) is false because from a point ON circle, exactly \(1\) tangent can be drawn, not \(2\).
Assertion (A) is true (no tangent can be drawn from inside). Reason (R) is false because from a point ON circle, exactly \(1\) tangent can be drawn, not \(2\).
Correct Answer: (C)
Assertion (A) is true. Reason (R) is false.
Assertion (A) is true. Reason (R) is false.
Assertion (A)
Distance between two parallel tangents of a circle of radius \(4\text{ cm}\) is \(8\text{ cm}\).
Reason (R)
Parallel tangents touch at endpoints of a diameter, so distance \(= 2r = 2 \times 4 = 8\text{ cm}\).
Correct Option: (B)
Both A and R are true.
Both A and R are true.
Correct Answer: (B)
Both A and R are true.
Both A and R are true.
Assertion (A)
Perimeter of \(\triangle PAB\) circumscribing a circle with tangents \(PQ, PR\) is \(2 \times PQ\).
Reason (R)
Tangents from an external point are equal (\(AX = AQ, BX = BR\)).
Correct Option: (B)
Both A and R are true and R provides equal tangent substitutions.
Both A and R are true and R provides equal tangent substitutions.
Correct Answer: (B)
Both A and R are true.
Both A and R are true.
Assertion (A)
If distance of point \(P\) from centre \(O\) is \(13\text{ cm}\) and radius is \(5\text{ cm}\), length of tangent \(PT = 12\text{ cm}\).
Reason (R)
In right \(\triangle OPT\), \(OT^2 = OP^2 + PT^2 \implies 13^2 = 5^2 + PT^2 \implies PT^2 = 169 - 25 = 144 \implies PT = 12\text{ cm}\).
Correct Option: (A)
Pythagoras theorem \(13^2 = 5^2 + PT^2 \implies PT = 12\text{ cm}\). Reason explains Assertion.
Pythagoras theorem \(13^2 = 5^2 + PT^2 \implies PT = 12\text{ cm}\). Reason explains Assertion.
Correct Answer: (A)
Reason explains Assertion.
Reason explains Assertion.
Assertion (A)
If \(PA\) and \(PB\) are tangents from external point \(P\) to circle with centre \(O\) such that \(\angle APB = 60^\circ\), then \(\angle AOB = 120^\circ\).
Reason (R)
In quadrilateral \(OAPB\), \(\angle OAP = \angle OBP = 90^\circ\), so \(\angle AOB + \angle APB = 180^\circ \implies \angle AOB = 180^\circ - 60^\circ = 120^\circ\).
Correct Option: (A)
Opposite angles in quadrilateral \(OAPB\) are supplementary: \(\angle AOB + \angle APB = 180^\circ\). Reason explains Assertion.
Opposite angles in quadrilateral \(OAPB\) are supplementary: \(\angle AOB + \angle APB = 180^\circ\). Reason explains Assertion.
Correct Answer: (A)
Reason explains Assertion.
Reason explains Assertion.
Assertion (A)
If a square is inscribed in a circle of radius \(r\), its side length is \(r\sqrt{2}\).
Reason (R)
The diagonal of square inscribed in circle is equal to radius \(r\).
Correct Option: (C)
Assertion (A) is true (diagonal \(= 2r\), side \(a = 2r/\sqrt{2} = r\sqrt{2}\)). Reason (R) is false because diagonal equals diameter \(2r\), not radius \(r\).
Assertion (A) is true (diagonal \(= 2r\), side \(a = 2r/\sqrt{2} = r\sqrt{2}\)). Reason (R) is false because diagonal equals diameter \(2r\), not radius \(r\).
Correct Answer: (C)
Assertion (A) is true. Reason (R) is false.
Assertion (A) is true. Reason (R) is false.
Assertion (A)
If \(PA\) and \(PB\) are tangents, \(\triangle PAB\) is always an equilateral triangle.
Reason (R)
Since \(PA = PB\), \(\triangle PAB\) is always an isosceles triangle, and becomes equilateral only when \(\angle APB = 60^\circ\).
Correct Option: (D)
Assertion (A) is false (it is always isosceles, equilateral only when angle is \(60^\circ\)). Reason (R) is true.
Assertion (A) is false (it is always isosceles, equilateral only when angle is \(60^\circ\)). Reason (R) is true.
Correct Answer: (D)
Assertion false. Reason (R) is true.
Assertion false. Reason (R) is true.
Assertion (A)
The line segment joining external point \(P\) to centre \(O\) bisects \(\angle APB\) between the two tangents.
Reason (R)
Tangents \(PA\) and \(PB\) drawn from an external point are equal in length.
Correct Option: (B)
Both A and R are true properties of tangents from an external point.
Both A and R are true properties of tangents from an external point.
Correct Answer: (B)
Both A and R are true.
Both A and R are true.
Assertion (A)
Number of tangents drawn from an external point to a circle is \(2\).
Reason (R)
Number of tangents from an internal point is \(1\).
Correct Option: (C)
Assertion (A) is true. Reason (R) is false because number of tangents from internal point is \(0\).
Assertion (A) is true. Reason (R) is false because number of tangents from internal point is \(0\).
Correct Answer: (C)
Assertion (A) is true. Reason (R) is false.
Assertion (A) is true. Reason (R) is false.
Assertion (A)
If \(TP\) and \(TQ\) are tangents to a circle with centre \(O\) such that \(\angle POQ = 110^\circ\), then \(\angle PTQ = 70^\circ\).
Reason (R)
Quadrilateral \(OPTQ\) has sum of interior angles \(360^\circ\).
Correct Option: (B)
Both A and R are true (\(360 - 90 - 90 - 110 = 70^\circ\)).
Both A and R are true (\(360 - 90 - 90 - 110 = 70^\circ\)).
Correct Answer: (B)
Both A and R are true.
Both A and R are true.
Assertion (A)
A circle has infinitely many tangents.
Reason (R)
A circle has infinitely many points on its circumference, and at each point a unique tangent can be drawn.
Correct Option: (B)
Both A and R are true.
Both A and R are true.
Correct Answer: (B)
Both A and R are true.
Both A and R are true.
Assertion (A)
From a point \(Q\), length of tangent is \(24\text{ cm}\) and distance from centre is \(25\text{ cm}\). Radius is \(7\text{ cm}\).
Reason (R)
In right triangle formed, \(\text{Radius}^2 = \text{Distance}^2 + \text{Tangent}^2\).
Correct Option: (C)
Assertion (A) is true (\(r = \sqrt{25^2 - 24^2} = 7\text{ cm}\)). Reason (R) is false because formula is \(\text{Distance}^2 = \text{Radius}^2 + \text{Tangent}^2\).
Assertion (A) is true (\(r = \sqrt{25^2 - 24^2} = 7\text{ cm}\)). Reason (R) is false because formula is \(\text{Distance}^2 = \text{Radius}^2 + \text{Tangent}^2\).
Correct Answer: (C)
Assertion (A) is true. Reason (R) is false.
Assertion (A) is true. Reason (R) is false.
Assertion (A)
Two concentric circles have radii \(5\text{ cm}\) and \(3\text{ cm}\). Length of chord of larger circle touching smaller circle is \(8\text{ cm}\).
Reason (R)
Radius \(r = 3\text{ cm} \perp\) chord. Half chord \(= \sqrt{5^2 - 3^2} = \sqrt{16} = 4\text{ cm}\). Total chord \(= 2 \times 4 = 8\text{ cm}\).
Correct Option: (A)
Pythagoras theorem gives half-chord \(= 4\), so full chord \(= 8\text{ cm}\). Reason explains Assertion.
Pythagoras theorem gives half-chord \(= 4\), so full chord \(= 8\text{ cm}\). Reason explains Assertion.
Correct Answer: (A)
Reason explains Assertion.
Reason explains Assertion.
Assertion (A)
If a parallelogram circumscribes a circle, it is a rhombus.
Reason (R)
Circumscribing quadrilateral gives \(AB + CD = AD + BC\). In a parallelogram, \(AB = CD\) and \(AD = BC \implies 2AB = 2AD \implies AB = AD\), so all sides are equal.
Correct Option: (A)
Proof that circumscribing parallelogram is a rhombus. Reason explains Assertion.
Proof that circumscribing parallelogram is a rhombus. Reason explains Assertion.
Correct Answer: (A)
Reason explains Assertion.
Reason explains Assertion.
Assertion (A)
The lengths of tangents drawn from an external point to a circle are equal (\(PA = PB\)).
Reason (R)
In right triangles \(\triangle OAP\) and \(\triangle OBP\), \(OA = OB\) (radii), \(OP = OP\) (common), \(\angle OAP = \angle OBP = 90^\circ\). By RHS, \(\triangle OAP \cong \triangle OBP \implies PA = PB\).
Correct Option: (A)
Theorem 10.2 proof via RHS congruence. Reason explains Assertion.
Theorem 10.2 proof via RHS congruence. Reason explains Assertion.
Correct Answer: (A)
Reason explains Assertion.
Reason explains Assertion.
Assertion (A)
A circle can have at most two parallel tangents along any given line.
Reason (R)
A secant intersects a circle in two distinct points.
Correct Option: (B)
Both A and R are true circle geometry facts.
Both A and R are true circle geometry facts.
Correct Answer: (B)
Both A and R are true.
Both A and R are true.
Assertion (A)
The point of contact of a tangent is the common point of the tangent line and the circle.
Reason (R)
A line intersecting a circle at two points is called a secant.
Correct Option: (B)
Both A and R are true definitions.
Both A and R are true definitions.
Correct Answer: (B)
Both A and R are true.
Both A and R are true.
Assertion (A)
If radii of two concentric circles are \(13\text{ cm}\) and \(5\text{ cm}\), length of chord of outer circle touching inner circle is \(24\text{ cm}\).
Reason (R)
Length of chord formula is \(\sqrt{R^2 - r^2}\).
Correct Option: (C)
Assertion (A) is true (\(2\sqrt{169-25} = 2(12) = 24\text{ cm}\)). Reason (R) is false because chord length is \(2\sqrt{R^2-r^2}\), missing factor \(2\).
Assertion (A) is true (\(2\sqrt{169-25} = 2(12) = 24\text{ cm}\)). Reason (R) is false because chord length is \(2\sqrt{R^2-r^2}\), missing factor \(2\).
Correct Answer: (C)
Assertion (A) is true. Reason (R) is false.
Assertion (A) is true. Reason (R) is false.
Assertion (A)
Tangents at endpoints of any chord are parallel.
Reason (R)
Tangents are parallel only when drawn at endpoints of a diameter.
Correct Option: (D)
Assertion (A) is false. Reason (R) is true.
Assertion (A) is false. Reason (R) is true.
Correct Answer: (D)
Assertion false. Reason (R) is true.
Assertion false. Reason (R) is true.
Assertion (A)
A circle can have an infinite number of parallel tangents along a given secant.
Reason (R)
For any given secant, there are at most two tangents parallel to it.
Correct Option: (D)
Assertion (A) is false (exactly two parallel tangents exist along a direction). Reason (R) is true.
Assertion (A) is false (exactly two parallel tangents exist along a direction). Reason (R) is true.
Correct Answer: (D)
Assertion false. Reason (R) is true.
Assertion false. Reason (R) is true.
Assertion (A)
Tangents drawn at the ends of a diameter of a circle are parallel.
Reason (R)
At endpoints \(A, B\) of diameter \(AB\), radii are perpendicular to tangents, making co-interior angles \(90^\circ + 90^\circ = 180^\circ\), which implies parallel lines.
Correct Option: (A)
Co-interior angles sum to \(180^\circ\), so lines are parallel. Reason explains Assertion.
Co-interior angles sum to \(180^\circ\), so lines are parallel. Reason explains Assertion.
Correct Answer: (A)
Reason explains Assertion.
Reason explains Assertion.
Assertion (A)
If \(OP = 5\text{ cm}\) and radius \(r = 5\text{ cm}\), length of tangent \(PT = 5\text{ cm}\).
Reason (R)
If \(OP = r\), point \(P\) lies on the circle, so length of tangent from \(P\) is \(0\).
Correct Option: (D)
Assertion (A) is false (length is \(0\)). Reason (R) is true.
Assertion (A) is false (length is \(0\)). Reason (R) is true.
Correct Answer: (D)
Assertion false. Reason (R) is true.
Assertion false. Reason (R) is true.
Assertion (A)
The angle between two tangents drawn from an external point is always acute.
Reason (R)
The angle between two tangents can be acute, right, or obtuse depending on distance from centre.
Correct Option: (D)
Assertion (A) is false (e.g. if \(OP = r\sqrt{2}\), angle is \(90^\circ\)). Reason (R) is true.
Assertion (A) is false (e.g. if \(OP = r\sqrt{2}\), angle is \(90^\circ\)). Reason (R) is true.
Correct Answer: (D)
Assertion false. Reason (R) is true.
Assertion false. Reason (R) is true.
Assertion (A)
In figure, if \(PA\) and \(PB\) are tangents with \(\angle APB = 80^\circ\), then \(\angle POA = 50^\circ\).
Reason (R)
\(\angle POA = \angle APB / 2 = 40^\circ\).
Correct Option: (C)
Assertion (A) is true (\(\angle APO = 40^\circ \implies \angle POA = 90 - 40 = 50^\circ\)). Reason (R) is false because \(\angle POA = 90^\circ - \angle APO = 50^\circ\), whereas \(\angle APO = 40^\circ\).
Assertion (A) is true (\(\angle APO = 40^\circ \implies \angle POA = 90 - 40 = 50^\circ\)). Reason (R) is false because \(\angle POA = 90^\circ - \angle APO = 50^\circ\), whereas \(\angle APO = 40^\circ\).
Correct Answer: (C)
Assertion (A) is true. Reason (R) is false.
Assertion (A) is true. Reason (R) is false.
Assertion (A)
The tangent at any point of a circle is perpendicular to the radius through point of contact.
Reason (R)
The radius \(OP\) is the shortest line segment from centre \(O\) to tangent line \(XY\), and shortest distance is perpendicular distance.
Correct Option: (A)
Theorem 10.1 proof: Shortest distance from a point to a line is perpendicular. Reason explains Assertion.
Theorem 10.1 proof: Shortest distance from a point to a line is perpendicular. Reason explains Assertion.
Correct Answer: (A)
Reason explains Assertion.
Reason explains Assertion.
Assertion (A)
If \(\angle AOB = 100^\circ\) for radii \(OA, OB\), angle between tangents at \(A, B\) is \(80^\circ\).
Reason (R)
Angle between radii and angle between tangents are supplementary.
Correct Option: (B)
Both A and R are true (\(180 - 100 = 80^\circ\)).
Both A and R are true (\(180 - 100 = 80^\circ\)).
Correct Answer: (B)
Both A and R are true.
Both A and R are true.