✓ Board-verified
Mathematics · 2025 · 3 marks
CBSE 2025 · Region 5 · Set 1 · Q31
In the adjoining figure, XY and X'Y' are parallel tangents to a circle with centre O. Another tangent AB touches the circle at C intersecting XY at A…
In the adjoining figure, XY and X'Y' are parallel tangents to a circle with centre O. Another tangent AB touches the circle at C intersecting XY at A and $\displaystyle \mathrm{X}^{\prime} \mathrm{Y}^{\prime}$ at B . Prove that AB subtends right angle at the centre of the circle; or $\displaystyle \angle \mathrm{AOB}=90^{\circ}$.
Marking-scheme solution
Join OC.
\[\begin{aligned}
& \triangle \mathrm{POA} \cong \triangle \mathrm{COA} \\
& \angle \mathrm{POA}=\angle \mathrm{COA}
\end{aligned}
\]
Similarly, \(\displaystyle \angle \mathrm{QOB}=\angle \mathrm{COB}\)
\[\angle \mathrm{POA}+\angle \mathrm{QOB}+\angle \mathrm{COA}+\angle \mathrm{COB}=180^{\circ}
\]
\[\begin{aligned}
& \Rightarrow 2(\angle \mathrm{COA}+\angle \mathrm{COB})=180^{\circ} \\
& \Rightarrow \angle \mathrm{COA}+\angle \mathrm{COB}=90^{\circ} \\
& \therefore \angle \mathrm{AOB}=90^{\circ}
\end{aligned}
\]
\[\begin{aligned}
& \triangle \mathrm{POA} \cong \triangle \mathrm{COA} \\
& \angle \mathrm{POA}=\angle \mathrm{COA}
\end{aligned}
\]
Similarly, \(\displaystyle \angle \mathrm{QOB}=\angle \mathrm{COB}\)
\[\angle \mathrm{POA}+\angle \mathrm{QOB}+\angle \mathrm{COA}+\angle \mathrm{COB}=180^{\circ}
\]
\[\begin{aligned}
& \Rightarrow 2(\angle \mathrm{COA}+\angle \mathrm{COB})=180^{\circ} \\
& \Rightarrow \angle \mathrm{COA}+\angle \mathrm{COB}=90^{\circ} \\
& \therefore \angle \mathrm{AOB}=90^{\circ}
\end{aligned}
\]More from Circles
- In the given figure, TA is a tangent to the circle with centre O such that OT=4 cm, ∠ OTA=30^°, then length of TA is:2023
- Assertion: Radius is the smallest distance of a tangent from the centre of the circle. Reason (R): Radius is perpendicular to the tangent.2026
- Assertion: The tangents drawn at the end points of a diameter of a circle, are parallel. Reason (R): Diameter of a circle is the longest chord.2024
- Prove that the parallelogram circumscribing a circle is a rhombus.2024
- A backyard is in the shape of a triangle ABC with right angle at B. AB=7 m and BC=15 m. A circular pit was dug inside it such that it touches the…2024
- PQ and PR are two tangents to a circle with centre O and radius 5 cm. AB is another tangent to the circle at C which lies on OP. If OP=13 cm, then…2026
- In Fig. 1, AB is diameter of a circle centered at O. BC is tangent to the circle at B. If OP bisects the chord AD and ∠ AOP=60^°, then find m ∠ C.…2022
- PQ is tangent to a circle with centre O. If ∠ POR=65^°, then m ∠ PTR is2026
CBSE Class 10 Mathematics past-paper question from the 2025 board exam, with the answer as CBSE’s own marking scheme gives it. Where our answers come from.