Two circles intersect at two points. Which of the following statements is true?
Only two common tangents exist
Step 1 – Recall the tangent-count rule for two circles:
The number of common tangents depends on the relative position of the two circles:
• Far apart (no intersection): 4 common tangents (2 direct + 2 transverse).
• Touching externally: 3 common tangents.
• Intersecting at two points: only the 2 direct (external) tangents exist; the transverse tangents disappear.
• Touching internally: 1 common tangent.
• One inside the other: 0 common tangents.
Step 2 – Apply this to the given case:
Two intersecting circles have exactly two common tangents.
Options 1, 3 and 4 are incorrect: there are no internal tangents to compare, the count does not depend on radii once intersection is given, and intersecting circles share two points — they cannot be concentric.
Hence the answer is Only two common tangents exist — option 2.
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Important Questions from Circles, Chords and Tangents
A chord 21 cm long is drawn in a circle of diameter 25 cm. The perpendicular distance of the chord from the centre is:
AB is a chord of a circle with centre O and P is any point on the circle. If ∠APB = 112°, then what is the measure of ∠OAB ?
In a circle with radius 5 cm, a chord is at a distance of 3 cm from the centre. The length of the chord is:
The distance between the centres of two circles is 24 cm. If the radius of the two circles are 4 cm and 8 cm, then what is the sum of the lengths (in cm) of the direct common tangent and the transverse common tangent?
An equilateral triangle ABC and a scalene triangle DBC are inscribed in a circle on same side of the arc. what is ∠BDC equal to?
