Question

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

Answer 1

GIVEN:
Radius = 2.5 cm
DISTANCE of tangent = 6 cm

From the figure we have,
OA = OB = 2.5 cm
AP = 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 + 36
x² + 5x - 36 = 0
x² +9x - 4x -36 = 0

[By factorization]

x(x +9) -4(x +9)=0
(x +9) (x -4) = 0
(x +9) = 0  or (x -4) = 0
x = -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.

HOPE THIS WILL HELP YOU...

Answer 2

Hello dear

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

Now, the hypotenuse OP equals
$\sqrt{ {2.5 }^{2} } + \sqrt{ {6}^{2} }$

=6.52.52+62=6.5, hence x=6.5−2.5=4x=6.5−2.5=4.