P is a point at a distance 12 cm from the centre O of a circle of radius 5 cm. PA and PB are the tangents to the circle A and B. If AP = 12 cm then find the value of chord AB
Answers
Given:
✰ P is a point at a distance 12 cm from the centre O of a circle of radius 5 cm.
✰ PA and PB are the tangents to the circle A and B.
✰ AP = 12 cm
To find:
✠ The value of chord AB.
Solution:
We know that,
⟹ d = 12 cm
⟹ r = 5 cm
Here,
- d is the point P at a distance 12 cm from the centre O.
- r is of a circle of 5 cm.
✭ Length of the tangent = l✭
✭ Length of the tangent = √(d²-r²) ✭
➤ l = √(12²-5²)
➤ l = √144-25
➤ l = √119
➤ PA = PB = √119 cm
✭ Area of rhombus = 1/2 × product of diagonals ✭
⟼ Area of ∆OAP = ∆OBP = 1/2 × 5 × √119
⟼ Area of rhombus OAPB = 2 × 1/2 × 5 × √119
⟼ Area of rhombus OAPB = 5√119
- One diagonal = OP = 12 cm
- Another diagonal = AB
Hence,
⟼ 1/2 × 12 × AB = 5√119
⟼ 6 × AB = 5√119
⟼ AB = 5√119/6 cm
∴ The value of chord AB = 5√119/6 cm
_______________________________
Given : P is a point at a distance 12 cm from the centre O of a circle of radius 5 cm.
PA and PB are the tangents to the circle at A and B.
To find : the value of chord AB.
Solution:
OP = 12 cm
OA = OB = Radius = 5 cm
PA² = OP² - OA²
=> PA² = 12² - 5² = 119
=> PA = √119
area of ΔOAP = ΔOBP = (1/2) * 5 * √119
=> area of OAPB ( rhombus) = 2 * (1/2) * 5 * √119 = 5√119
Also area of rhombus = (1/2) * product of diagonals
one diagonal = OP = 12
another diagonal = AB
Hence = (1/2) * 12 * AB = 5√119
=> AB = 5√119/6 cm
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