x2
The line x cos a + y sin a = p will be tangent to the conic
+
کل
= 1 if
a 2
b2
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Answer:
Answer
xcosα+ysinα=p
⇒y=
sinα
1
2
−xcosα
⟶(i)
⇒
a
2
x
2
+
b
2
y
2
=1
b
2
x
2
+a
2
y
2
=a
2
b
2
b
2
x
2
+a
2
(pcscα−xcotα)
2
=a
2
b
2
⇒x
2
[b
2
+a
2
cot
2
α]−2a
2
(pcscαcotα)x+a
2
(p
2
csc
2
x−6
2
)=0→(ii)
If xcosα+ysinα=p is tangent to
a
2
x
2
+
b
2
y
2
=1, then equation (ii) has equal roots.
∴4a
4
p
2
csc
2
α−4(b
2
+0
2
cot
2
α)(a
2
p
2
csc
2
α−a
2
b
2
)=0
⇒b
2
p
2
csc
2
α−b
4
−a
2
b
2
cot
2
α=0
⇒p
2
−b
2
sin
2
α−a
2
cos
2
α=0
⇒p
2
=a
2
cos
2
α+b
2
sin
2
α
Hence, solved.
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