Math, asked by keerthikamereddy, 4 months ago

2.
Show that the condition that the tangents drawn
from (0,0) to S= x2 + y2 + 2gx +2 fy+c=0
be perpendicular to each other is g? + f = 20
(MAY-2016

Answers

Answered by rose525
3

Answer:

Given circle =x

2

+y

2

+2gx+2fy+c=0

Let LM be the chord

of circle C and subtend right angle at (0,0)O,P(h,k) be fool of ⊥ from O on chord LM

as OP⊥ML, eqn of LM is:

y−k=

k

h

(x−h)

hx+ky=h

2

+k

2

.......... (1)

equation of pair of line joining point of intersection of (1) with S=0⇒x

2

+y

2

+2(gx+fy)

h

2

+k

2

(hx+ky

+c

(h

2

+k

2

)

2

(hx+ky)

2

=0 ........... (2)

as lines in (2) are at right angles

1+1=

h

2

+k

2

2gh+2fk

+c

(h

2

+k

2

)

(h

2

+k

2

)

=0

h

2

+k

2

+gh+fk+

2

c

=0

∴ Locus of (h,k) is ⇒S=x

2

+y

2

+gx+fy+

2

c

=0

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