Find the equation of the ellipse satisfying the given condition e=34, foci on Y-axis, centre at origin and passes through (6,4).
Find the equation of the hyperbola with vertices at (±5,0) and foci (±7,0)
Let the equation of ellipse be x2a2+y2b2=1 …(i)
If the ellipse passes through (6,4) then
We know that, e=√ [∵b>a]
⇒ 34=√ ⇒ 916=1−a2b2
⇒ a2b2=1−916=716 ⇒ a2=716b21 …(iii)
From Eqs. (i) and (iii), we get
36×167b2+16b2=1 ⇒ 167b2(36+7)=1
⇒1b2=716×43 ⇒ b2=16×437
Put the value of b^2 in Eq. (iii) we get
On substituting the values of a2 and b2 in Eq. (i), we get
∴ 16x2+7y2=688, which is the required equation of ellipse.
Since, the vertices are on the X-axis with origin at the mid-point.
Then, equation of the hyperbola is of the form
Since, the vertices are (±5,0) and foci are (±7,0])
∴ a=5 and c=7
We know that, in hyperbola
On putting the value of b2 in Eq. (i) we get