Geometry

By the tangent-radius theorem, XU is perpendicular to OU at point U. Triangle XUO is right-angled at U. Using Pythagoras: OX² = XU² + OU² = 15² + 8² = 225 + 64 = 289. Therefore OX = √289 = 17 cm.

When a chord subtends 180° at the center, it is a diameter. According to the theorem: 'The angle subtended by a chord at the center is twice the angle subtended at the circumference on the same side.' Therefore, if the central angle is 180°, the angle at the circumference = 180°/2 = 90°. This is a fundamental property of circles.

Let the angle at the centre be C and at the major arc be M. Given: C + M = 180°. For any chord, the angle at the centre is twice the angle at the circumference on the same side: C = 2M. Substituting: 2M + M = 180°, so 3M = 180°, M = 60°. Therefore, C = 180° - 60° = 120°. The angle at the centre is 120°.

For two circles touching externally at P, AB is a direct common tangent with A and B as points of contact. The radius to the point of tangency is perpendicular to the tangent, so OA is perpendicular to AB and OB is perpendicular to AB. This means OA and OB are parallel. Given ∠PAB = 40°, the line PA makes a 40° angle with AB. Since OA is perpendicular to AB, ∠PAO = 90° - 40° = 50°. Triangle PAO is isosceles (OA = OP as radii), so ∠ABP = ∠PAO = 50°. Alternatively, using properties of tangents and radii, the angle between the tangent and chord equals the angle in the alternate segment.

The length of a chord in a circle of radius r that subtends an angle θ at the center is given by: chord length = 2r sin(θ/2). Here r = 10 cm and θ = 60° (assuming 600 is a typo for 60°). Chord length = 2 × 10 × sin(60°/2) = 20 × sin(30°) = 20 × 0.5 = 10 cm. If the angle is literally 600°, then chord length = 20 × sin(300°) = 20 × (-√3/2) which is negative and doesn't make physical sense. Assuming standard geometric context, the angle is 60°, making the chord length 10 cm.

For chord length L in a circle of radius R, the perpendicular distance from center to chord is sqrt(R² - (L/2)²). For 42 cm chord: d1 = sqrt(29² - 21²) = sqrt(841 - 441) = sqrt(400) = 20 cm. For 40 cm chord: d2 = sqrt(29² - 20²) = sqrt(841 - 400) = sqrt(441) = 21 cm. The chords are parallel and on opposite sides of the center (since the question asks for minimum distance). Minimum separation = 20 + 21 = 41 cm. But wait - the answer options are small (1-4 cm), suggesting these chords are on the SAME side of center. Then minimum distance between them = |21 - 20| = 1 cm. Answer is 1 cm (Option B).

Tangent PQ is perpendicular to radius OQ at point Q, forming right triangle ΔOQP with OQ = 10 cm and PQ = 24 cm. By Pythagoras theorem: OP² = OQ² + PQ² = 100 + 576 = 676, so OP = 26 cm. Option C (13 cm) is half the correct answer.

By the tangent theorem: tangent² + radius² = distance from center². Given tangent = 15 cm, distance = 17 cm. So 15² + r² = 17², which gives 225 + r² = 289, therefore r² = 64 and r = 8 cm.

Since L is the circumcentre, it's equidistant from all vertices. LM is the perpendicular bisector of YZ, so LY = LZ and triangles LYM and LZM are congruent right triangles. Angle YLZ (central angle) = 2 × angle YXZ = 2 × 60° = 120°. Therefore angle YLM = 120°/2 = 60°. 120° is the full central angle, 90° would be for orthocentre properties, and 180° is a straight angle.

For two circles with radii r1=8 and r2=6 touching externally, the direct common tangent length = 2√(r1×r2) = 2√(8×6) = 2√48 = 2×6.928 = 13.856 cm. This formula comes from the right triangle formed by the tangent line and lines to centers.

In a circle, an exterior angle formed by extending a chord equals half the difference of 360° and the central angle subtended by the intercepted arc. The arc ABC subtends 132° at center O, so the remaining arc AC measures 360° - 132° = 228°. The exterior angle ∠CBP equals half of this remaining arc: 228°/2 = 114°. However, ∠CBP is the angle formed outside the circle, which is supplementary to the angle in the alternate segment. Using the exterior angle property for chords: ∠CBP = (360° - 132°)/2 = 66°. Alternatively, ∠CBP equals the angle in the opposite segment, which is half the central angle subtended by the same arc: 132°/2 = 66°.

In a triangle, the centroid divides each median in a 2:1 ratio, with the longer segment between the vertex and centroid. Since BG = 12 cm (vertex to centroid), the shorter segment GE = 6 cm. Therefore the total median BE = BG + GE = 12 + 6 = 18 cm.

When an angle subtended by a chord at the center of a circle is 180°, the chord must be a diameter of the circle. This is because the angle at the center is twice the angle at the circumference, and a 180° central angle means the arc is a semicircle. Therefore, if chord PQ is 54 cm, the diameter is also 54 cm.

Given diameter = 20 cm, so radius r = 10 cm. The perpendicular distance from the center to chord OP is 6 cm. Using the formula: (half chord)² + (distance from center)² = radius². So half-chord = √(10² - 6²) = √(100-36) = √64 = 8 cm. Full chord = 16 cm. Sum of radius and chord = 10 + 16 = 26 cm.

In a right triangle, the tangent is perpendicular to the radius at the point of contact. Using Pythagoras: PX² + OP² = OX², so PX = √(26² - 10²) = √(676 - 100) = √576 = 24 cm.