C
find the mid points with the help of points A (x,y) and
B (x² y²)
Answers
Answer:
x2+y
2
=a
2
......(1)
hx+ky=h
2
+k
2
.....(2)
where (h,k) is middle point of chord AB, by T=S
1
.
AP and PB are perpendiculars as AB subtends an angle of 90
∘
at P(α,β).
If A and B be (x
1
,y
1
) and (x
2
,y
2
) respectively
x
1
−α
y
1
−β
⋅
x
2
−α
y
2
−β
=−1
or (x
1
−α)(x
1
−α)+(y
1
−β)(y
2
−β)=0
or (x
1
x
2
+y
1
y
2
)−α(x
1
+x
2
)−β(y
1
+y
2
)+α
2
+β
2
=0.....(3)
Eliminating y between (1) and (2), we will get a quadratic
x
2
(h
2
+k
2
)−2λhx+λ
2
−a
2
k
2
=0
λx
2
−2λhx+λ
2
−a
2
k
2
=0
Similarly λy
2
−2λky+λ
2
−a
2
h
2
=0
Putting the values of x
1
x
2
,y
1
y
2
,x
1
+x
2
,y
1
+y
2
from above in (3), we get
λ
λ
2
−a
2
k
2
+λ
2
−a
2
h
2
−α.2h−β.2k+α
2
+β
2
=0
or 2(λ−hα−kβ)+(α
2
+β
2
)−a
2
=0
Dividing by 2 and putting λ=h
2
+k
2
and then generalizing
x
2
+y
2
−αx−βy+
2
1
(α
2
+β
2
−a
2
)=
Step-by-step explanation:
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