Question

find the power of P(x1,y1) with respect to the circle S = x² + y² + 2gx + 2fy + c = 0​

Answer 1

Answer:

The length of tangent from (2,−3) to a circle x

2

+y

2

=1 is

Step-by-step explanation:

Let S=0 i.e. x

2

+y

2

+2gx+2gx+2fy+c=0 be the equation of circle $$P(x_1 y_1)$ be the external point and PT & PS are 2 tangents to give circle.

Since, ΔPTC is a right angled triangle.

⇒PT

2

+TC

2

=PC

2

__(A)

But TC = radius of circle =

g

2

+f

2

−c

⇒TC

2

=g

2

+f

2

−c __(I)

Also, PC

2

=[x

1

−(g)]

2

+[y

1

−(−f)]

2

PC

2

=(x

1

+g)

2

+(y

1

+f)

2

__(II)

Using (I) & (II) in equation A, we get

PT

2

+g

2

+f

2

−C=(x

1

+g)

2

+(y

1

+f)

2

⇒PT

2

+g

2

+f

2

−c=x

1

2

+g

2

+2gx

1

+g

2

+y

1

2

+2fy

1

+f

2

⇒PT

2

=x

1

2

+y

1

2

+2gx

1

+2fy

1

+c

⇒∣PT∣=

x

1

2

+y

1

2

=2gx

1

+2fy

1

+c

=

s

11

Hence proved

solution