Fsc Part 2 Mathematics (Complete Solutions)

Q7 Show that the parametric equations:

(i)     x = at2 , y = 2at represent the equation of parabola y2 = 4ax
(ii)     x = acosθ , y = bsinθ represent the equation of ellipse
 x2 a2
+
 y2 b2
= 1
(iii)     x = asecθ , y = btanθ represent the equation of hyperbola
 x2 a2
+
 y2 b2
= 1

Solution

(i)     x = at2 , y = 2at represent the equation of parabola y2 = 4ax
The parametric equations are:
x = at2        ..........(i)
y = 2at       ..........(ii)
From (ii)       y = 2at
By taking square on both sides
y2 = (2at)2
y2 = 4a2t2
y2 = 4a(at2)
So according to (i) x = at2
y2 = 4ax    which is the equation of parabola

(ii)     x = acosθ , y = bsinθ represent the equation of ellipse
 x2 a2
+
 y2 b2
= 1
The parametric equations are:
x = acosθ        ..........(i)
y = bsinθ       ..........(ii)
From (i)       x = acosθ

 x a
= cosθ
By taking square on both sides
(
 x a
)2 = cos2θ        ..........(a)
From (ii)      y = bsinθ

 y b
= sinθ
By taking square on both sides
(
 y b
)2 = sin2θ        ..........(b)
Now by adding equation (a) and (b)
(
 x a
)2  +  (
 y b
)2  = cos2θ + sin2θ
As cos2θ + sin2θ = 1
 x2 a2
+
 y2 b2
= 1    which is the equation of ellipse

(iii)     x = asecθ , y = btanθ represent the equation of hyperbola
 x2 a2
+
 y2 b2
= 1
The parametric equations are:
x = asecθ        ..........(i)
y = btanθ       ..........(ii)
From (i)       x = asecθ

 x a
= secθ
By taking square on both sides
(
 x a
)2 = sec2θ        ..........(a)
From (ii)      y = btanθ

 y b
= tanθ
By taking square on both sides
(
 y b
)2 = tan2θ        ..........(b)
Now by subtracting equation (a) and (b)
(
 x a
)2  -  (
 y b
)2  = sec2θ - tan2θ
As sec2θ - tan2θ = 1
 x2 a2
+
 y2 b2
= 1    which is the equation of ellipse
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