Circle and tangents
The tangent at ( 1 , 2 ) to the circle C 1 : x 2 + y 2 = 5 intersects the circle C 2 : x 2 + y 2 = 9 at A and B ; and the tangents at A and B on C 2 meet at P . If the coordinate of P is in the form ( b a , d c ) , where a , b are positive coprime integers, and c , d are positive coprime integers, then find a + b + c + d .
The answer is 37.
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2 solutions
same method !!! just compare the tangent and chord of contact's equation ..
Let T=(1,2), and O=(0.0).
TO is perpendicular bisector of AB, the chord of
C
2
.
PAB also is an isosceles triangle. So PT is also perpendicular bisector of AB.
So O-T-P is a st. line.
B
O
=
9
,
T
O
=
5
.
.
So right tangles PTB ~ OTB.
∴
T
O
B
O
=
B
O
P
O
,
⟹
P
O
=
5
9
.
T
O
P
O
=
5
5
9
=
5
9
.
∴
P
=
(
5
9
∗
1
,
5
9
∗
2
)
=
(
5
9
,
5
1
8
)
=
(
b
a
,
d
c
)
.
a
+
b
+
c
+
d
=
9
+
5
+
1
8
+
5
=
3
7
Equation of tangent at P Is T = 0
So equation of tangent x+2y = 5 1...
Let point P is (x1,y1)
Equation of chord of contact from (x1,y1) to circle x^2 + y^2 = 9 is
xx1+yy1 = 9 2.....
1 and 2 are same equations so
1/x1 = 2/y1 = 5/9
as simple as that!