SECTION B — Very Short Answer Type Questions [2 Marks Each]
- 2 Marks Q1. A tangent $PQ$ at a point $P$ of a circle of radius $5\text{ cm}$ meets a line through the center $O$ at a point $Q$ so that $OQ = 12\text{ cm}$. Find the length of $PQ$.
- 2 Marks Q2. Prove that the tangents drawn at the ends of a diameter of a circle are parallel.
- 2 Marks Q3. If tangents $PA$ and $PB$ from a point $P$ to a circle with center $O$ are inclined to each other at an angle of $80^\circ$, then find $\angle POA$.
- 2 Marks Q4. Prove that the angle between the two tangents drawn from an external point to a circle is supplementary to the angle subtended by the line-segment joining the points of contact at the center.
- 2 Marks Q5. A quadrilateral $ABCD$ is drawn to circumscribe a circle. Prove that $AB + CD = AD + BC$.
- 2 Marks Q6. Prove that the parallelogram circumscribing a circle is a rhombus.
- 2 Marks Q7. From a point $Q$, the length of the tangent to a circle is $24\text{ cm}$ and the distance of $Q$ from the center is $25\text{ cm}$. Find the radius of the circle.
- 2 Marks Q8. Two concentric circles are of radii $5\text{ cm}$ and $3\text{ cm}$. Find the length of the chord of the larger circle which touches the smaller circle
- 2 Marks Q9. $TP$ and $TQ$ are the two tangents to a circle with center $O$ so that $\angle POQ = 110^\circ$. Find $\angle PTQ$.
- 2 Marks Q10. If tangents $PA$ and $PB$ from a point $P$ to a circle with center $O$ are inclined to each other at angle $\theta$, show that $\sin(\frac{\theta}{2}) = \frac{r}{OP}$ where $r$ is the radius
- 2 Marks Q11. Prove that at any point on a circle, the tangent is perpendicular to the radius through the point of contact.
- 2 Marks Q12. A circle is inscribed in a $\triangle ABC$ having sides $8\text{ cm}$, $10\text{ cm}$, and $12\text{ cm}$. If the radius of the circle is $2\text{ cm}$, find the area of $\triangle ABC$ using the perimeter-inradius relation (or verify the tangent lengths).
- 2 Marks Q13. Prove that the lengths of tangents drawn from an external point to a circle are equal.
- 2 Marks Q14. If $PA$ and $PB$ are tangents from an external point $P$ to a circle with center $O$, and $PA = 8\text{ cm}$, find the length of $PB$
- 2 Marks Q15. In two concentric circles, a chord of length $8\text{ cm}$ of the larger circle touches the smaller circle. If the radius of the larger circle is $5\text{ cm}$, find the radius of the smaller circle
- 2 Marks Q16. A point $P$ is at a distance of $10\text{ cm}$ from the center of a circle. The length of the tangent drawn from $P$ to the circle is $8\text{ cm}$. Find the radius of the circle.
- 2 Marks Q17. If a circle touches all the four sides of a parallelogram $ABCD$, show that the parallelogram is a rhombus.
- 2 Marks Q18. Prove that the intercept of a tangent between two parallel tangents to a circle subtends a right angle at the center.
- 2 Marks Q19. Two circles touch each other externally at point $C$. $AB$ is a direct common tangent to the two circles. Prove that the angle formed by the tangent at the point of contact is $90^\circ$
- 2 Marks Q20. In $\triangle ABC$, if the inradius is $4\text{ cm}$ and the segments into which one side is divided by the point of contact are $6\text{ cm}$ and $8\text{ cm}$, find the other two sides.
- 2 Marks Q21. Find the distance between two parallel tangents of a circle of radius $4\text{ cm}$
- 2 Marks Q22. A circle touches the side $BC$ of $\triangle ABC$ at $P$ and produced sides $AB$ and $AC$ at $Q$ and $R$. Prove that $AQ = \frac{1}{2}(\text{perimeter of }\triangle ABC)$.
- 2 Marks Q23. If an isosceles triangle $ABC$ in which $AB = AC = 6\text{ cm}$ is inscribed in a circle of radius $5\text{ cm}$, find the distance of the chord $BC$ from the center.
- 2 Marks Q24. Prove that the angle between two tangents drawn from an external point is bisected by the line joining the external point to the center
- 2 Marks Q25. If the angle between two tangents drawn from an external point $P$ to a circle of radius $r$ and center $O$ is $60^\circ$, find the length of $OP$.
Academic Repository Disclaimer & எச்சரிக்கை அறிவிப்பு :
English: These model solutions are structured to global academic standards for self-assessment and practice purposes. Students are advised to verify with official textbooks for board examinations.
தமிழ்: இந்த மாதிரி கணிதத் தீர்வுகள் மாணவர்களின் சுய கற்றல் மற்றும் பயிற்சித் திறனுக்காக உலகளாவிய கல்வித் தரத்தில் உருவாக்கப்பட்டுள்ளன. தேர்வுகளுக்குத் தயாராகும் போது உங்கள் அதிகாரப்பூர்வ பாடப்புத்தகத்துடன் சரிபார்க்கவும்.
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