Let (x, y) be the coordinates of a point P in the plane with reference to the two perpendicular axes OX and OY. Let O ′ ( α, β ) be a point of the plane.
(I) If origin is shifted to O ′ ( α, β ) with O ′ X ′ and O ′ Y ′ as new set of axes where O ′ X ′ parallel to OX and O ′ Y ′ parallel to OY, then the new coordinates (x ′ , y ′ ) of P will be given by x ′ = x – α , y ′ = y – β
(II) If the origin is not shifted but axes are rotated about O through the angle θ in anticlockwise direction, then 
and 
(i) If the origin is not shifted, but both the axes are rotated about the origin in the anticlockwise direction through an angle of 45°, the equation x 2 – y 2 = a 2 attains the new form
Text Solution
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Ans.
(i)
Sol. x = 
– 
and y = 
+ 
so x 2 – y 2 = a 2 becomes ⇒ 
(x ′ – y ′ ) 2 – 
(x ′ + y ′ ) 2 = a 2 ⇒ – 2x ′ y ′ = a
2 ⇒ 2xy + a 2 = 0
(ii)
Sol. Using x = x 1 cos θ – y 1 sin θ ,
y = x 1 sin θ + y 1 cos θ
7x 2 +4xy+3y 2 =7(x 1 cos θ – y 1 sin θ ) 2 + 4(x 1 cos θ – y 1 sin θ )
(x 1 sin θ +y 1 cos θ )+ 3(x 1 sin θ + y 1 cos θ ) 2 If this is the form of ax
2 + by 2 then coefficient of xy = 0
⇒ –14 sin θ cos θ + 4 cos 2 θ – 4 sin 2 θ + 6 sin θ cos θ = 0
⇒ tan 2 θ = 1 ⇒ θ = π /8
(iii)
Sol. If origin is shifted to ( α, β ) without rotating the axis the new coordinates (x ′ , y ′ ) are given by
x ′ = x – α , y ′ = y – β here α = 1, β = –1
So (1, –2) becomes (1 – 1, –2 + 1) i.e. (0, –1)
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