Question

CBSE CLASS X

CIRCLES - MATHEMATICS

In the given figure, PT is a tangent and PAB is a Secant. If PT = 6cm and AB = 5cm, then find the length of PA.

Answer 1

Hi dear !!!

here is your answer !!!

from circle property we know

PT² = PA × PB

36 = PA ( PA + AB )

36 = PA² + 5 PA

PA² + 5 PA -36 = 0

USING SRIDHARACHARYA FORMULA

-b ± √ b² - 4ac / 2a

=> -5 ± √ 25 - 4× 36 × 1 / 2

=> 13-5 /2

on neglecting the negative value .

we get only one solution

=> 4cm

hence Length of PA is 4

hope it help you dear

thanks !!

Answer 2

Let PA be x .

Information given :-

Given ,PT = 6 cm

           PA = x [ I assumed it as x ]

           AB = 5 cm

From figure : PB = PA + AB

                  ==> PB = x + 5

Theorem

We know by properties of tangent that :

When a tangent and a chord meet externally or internally point then , the product of the line segments is equal to the square of the tangent !

This means that we can say :

PT² = PA × PB [ Using the above theorem ]

Length of PA

By using the formula :

( 6  )² = x ( x + 5 )

==> x ( x + 5 ) = 36

==> x² + 5 x = 36

==> x² + 5 x - 36 = 0

==> x² + 9 x - 4 x - 36 = 0

==> x ( x + 9 ) - 4 ( x + 9 ) = 0

==> ( x - 4 )( x + 9) = 0

The value of x

Either :

x = 4

or :

x = -9

The value of x cannot be negative as length cannot be negative.

So the value of x is 4.

$\underline{\underline{{\bf{ANSWER}}}}$

$\huge\boxed{\bf{\red{PA\:=\:4\:cm}}}$

Hope it helps u :-)

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