Question

The length of tangent to a circle of radius 2.5cm from an external point p is 6cm. find the distance is p from the nearest point of the circle

Answer 1

NIkita Mam has well answered here..

GIVEN:Radius = 2.5 cmDISTANCE of tangent = 6 cm
From the figure we have,OA = OB = 2.5 cmAP = 6 cm
Let BP = x cm
We know that the tangent at any point of a circle is perpendicular to the radius through the point of contact.
In right angled ∆OAP, OA ⟂ AP,OP² = OA² + AP²
[By Pythagoras theorem](2.5 +x)² = (2.5)² + 6²    [OP = OA + BP]6.25 + x² + 5x = 6.25 + 36x² + 5x - 36 = 0x² +9x - 4x -36 = 0
[By factorization]
x(x +9) -4(x +9)=0(x +9) (x -4) = 0(x +9) = 0  or (x -4) = 0x = -9 , x = 4 
Sides Can't be negative , so x= 4
Hence, the distance of P from the nearest point of the circle B is 4 cm.


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Answer 2

Here is your answer
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The shortest distance from point P to the circle is the length of x.


Hypotenuse = √ 2.5 ² + √ 6²

=>


=> 6.52.52+62=6.5

=> X =6.5−2.5

=> 4x = 6.5−2.5=4.