Plane Geometry
- A and B are centres of two circles of radii 11 cm and 6 cm, respectively. PQ is a direct common tangent to the circles. If AB = 13 cm, then length of PQ will be
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As per the given in question , we draw a figure of two circles of radii 11 cm and 6 cm
Here , AB = 13 cm
PQ = √AB² - (r1 - r2)²
PQ = √13² - (11 - 6)²Correct Option: C
As per the given in question , we draw a figure of two circles of radii 11 cm and 6 cm
Here , AB = 13 cm
PQ = √AB² - (r1 - r2)²
PQ = √13² - (11 - 6)²
PQ = √13² - 5² = √169 - 25
PQ = √144 = 12 cm.
- A point Q is 13 cm from the centre of a circle. The length of the tangent drawn from Q to a circle is 12 cm. The distance of Q from the nearest point of the circle is
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According to question , we draw a figure of a circle with centre O
Here ,QB = Tangent = 12 cm.
OQ = 13 cm.
∠QBO = 90°
From ∆ OQB,
OB = √OQ² - QB²
OB = √13² - 12² = √169 - 144Correct Option: B
According to question , we draw a figure of a circle with centre O
Here ,QB = Tangent = 12 cm.
OQ = 13 cm.
∠QBO = 90°
From ∆ OQB,
OB = √OQ² - QB²
OB = √13² - 12² = √169 - 144
OA = OB = √25 = 5 cm.
∴ AQ = Shortest distance = OQ – OA = 13 – 5 = 8 cm.
- AC is transverse common tangent to two circles with centres P and Q and radii 6 cm and 3 cm at the point A and C respectively. If AC cuts PQ at the point B and AB = 8cm then the length of PQ is :
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On the basis of question we draw a figure of two circles with centres P and Q and radii 6 cm and 3 cm at the point A and C respectively ,
In ∆ APB and ∆ BCQ,
∠ PAB = ∠ BCQ = 90°
∠ PBA = ∠ QBC
By AA – similarity,
∆ APB ~ ∆ BQC∴ AB = AP BC QC ⇒ 8 = 6 BC 3
Correct Option: D
On the basis of question we draw a figure of two circles with centres P and Q and radii 6 cm and 3 cm at the point A and C respectively ,
In ∆ APB and ∆ BCQ,
∠ PAB = ∠ BCQ = 90°
∠ PBA = ∠ QBC
By AA – similarity,
∆ APB ~ ∆ BQC∴ AB = AP BC QC ⇒ 8 = 6 BC 3 ⇒ BC = 8 × 3 = 4 cm. 6
∴ PQ = √AC² + (r1 + r2)²
PQ = √(8 + 4)² + (6 + 3)²
PQ = √12² + 9² = √144 + 81
PQ = √225 = 15 cm.
- XY and XZ are tangents to a circle, ST is another tangent to the circle at the point R on the circle, which intersects XY and XZ at S and T respectively. If XY = 15 cm and TX = 9 cm, then RT is
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As per the given in question , we draw a figure of a circle in which XY and XZ are tangents
We can say that the tangents drawn from an exterior point to a circle are equal.
∴ XY = XZ = 15 cm.
∴ TZ = XZ – TXCorrect Option: C
As per the given in question , we draw a figure of a circle in which XY and XZ are tangents
We can say that the tangents drawn from an exterior point to a circle are equal.
∴ XY = XZ = 15 cm.
∴ TZ = XZ – TX
TZ = 15 – 9 = 6 cm
Argain, TR = TZ = 6 cm
- A tangent is drawn to a circle of radius 6 cm from a point situated at a distance of 10 cm from the centre of the circle. The length of the tangent will be
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On the basis of question we draw a figure of a circle with centre O
AB is a tangent. OB is radius of circle.
OB ⊥ AB
OB = 6 cm.
OA = 10 cm.Correct Option: D
On the basis of question we draw a figure of a circle with centre O
AB is a tangent. OB is radius of circle.
OB ⊥ AB
OB = 6 cm.
OA = 10 cm.
∴ AB = √OA² - OB²
AB = √10² - 6² = √100 - 36
AB = √64 = 8 cm.
