Question:
If p = X cos θ – Y sin θ, q = X sin θ + Y cos θ and p2 + 4pq + q2 = AX2 + BY2 , 0 ≤ θ ≤ (π)/(2) . What is the value of θ ?
If p = X cos θ – Y sin θ, q = X sin θ + Y cos θ and p2 + 4pq + q2 = AX2 + BY2 , 0 ≤ θ ≤ (π)/(2) . What is the value of θ ?
Options:
- pi/6
- pi/2
- pi/3
- pi/4
- pi/6
- pi/2
- pi/3
- pi/4
Correct Answer: a)
Explanation:
‘ We ha√e,
p= Xcosf—Ysinb ..(i)
i q= Xsinf + Ycos# …(ii)
‘ and
2 +4pq + = AX? + BY? .. (iii)
i From Egs. (i) and (ii), we get
‘ p2+q2 = (X cos — Yin)? +(X sin + Y cos 0)? ‘
= pPrgt=x2+Y?
i and pg = (X cos — Y’sinf) (Xsin + Y cos 6)
= pg = (x? -Y?) sin cos 6 +XY cos 20
! 2 2_42.2 !
Sp t4pgtqt=X”+Y +2
(X2—Y?)sin20+ 4XY cos 20 “
‘ Gi√en, p? +4pq + ¢? =x? + BY?
XYZ eax? o√?
sin20+4XYcos20 = AX? +2BY?
‘ Coefficient of XY = 0 !
s.c0820=0=20=7%
a
1 f=1 I
hon a
H 44. 1
1 © = 1
‘
‘ We ha√e,
p= Xcosf—Ysinb ..(i)
i q= Xsinf + Ycos# …(ii)
‘ and
2 +4pq + = AX? + BY? .. (iii)
i From Egs. (i) and (ii), we get
‘ p2+q2 = (X cos — Yin)? +(X sin + Y cos 0)? ‘
= pPrgt=x2+Y?
i and pg = (X cos — Y’sinf) (Xsin + Y cos 6)
= pg = (x? -Y?) sin cos 6 +XY cos 20
! 2 2_42.2 !
Sp t4pgtqt=X”+Y +2
(X2—Y?)sin20+ 4XY cos 20 “
‘ Gi√en, p? +4pq + ¢? =x? + BY?
XYZ eax? o√?
sin20+4XYcos20 = AX? +2BY?
‘ Coefficient of XY = 0 !
s.c0820=0=20=7%
a
1 f=1 I
hon a
H 44. 1
1 © = 1
‘