find the equation of ellipse given that centre is (4,-1) focus is (1,-1) and passing through (8,0)
Answers
Step-by-step explanation:
Let C be the centre and F and F' be the foci of the ellipse.
Given F(1,−1),C=(4,−1)
F
′
(x,y) (say)
C is the Mid-point of FF',
4=
2
x+1
,−1=
2
y−1
⇒x=7,y=−1
F
′
(7,−1)
Since, the y-coordinates of F and F' are equal therefore, the transverse axis of the ellipse is parallel to x-axis.
The equation of the ellipse with centre at (4, -1) is
a
2
(x−4)
2
+
b
2
(y+1)
2
=1 ......... (i)
∣CF∣=
(4−1)
2
+(−1+1)
2
∣CF∣=3
⇒ae=3
⇒a
2
e
2
=9
Now, b
2
=a
2
−a
2
e
2
⇒b
2
=a
2
−9 ......... (ii)
As ellipse (i) passes through (8, 0),
a
2
16
+
b
2
1
=1
⇒16b
2
+a
2
=a
2
b
2
........ (iii)
16(a
2
−9)+a
2
(a
2
−9)
⇒a
4
−16a
2
+144=0
⇒a
4
−18a
2
−8a
2
+144=0
⇒(a
2
−18)(a
2
−8)=0
⇒a
2
=18,a
2
=8
a
2
=8 is inadmissible as it yields b
2
a negative value.
∴a
2
=18
∴ From (ii), b
2
=9
∴ Equation of the ellipse is
18
(x−4)
2
+
9
(y+1)
2
=1
⇒(x−4)
2
+(y+1)
2
=18
x
2
−8x+16+2y
2
+4y+2=18
⇒x
2
+2y
2
−8x+4y=0