7.3 Ellipses

Find the ellipses given the following conditions:

(a) The foci are and the length of the minor axis is 6.

(b) The foci are [math](2,3), (-4,3)[/math] and the length of the major axis is 8.

If the focal length of the ellipse [math](x-1)^2+4(y+1)^2=9[/math] is [math]\sqrt{a}[/math] where [math]a[/math] is an integer, what is the value of [math]a[/math]?

Find one focus of the ellipse [math]x^2+2y^2+2x-4y=-1[/math]

If the focal length of the ellipse [math]144-16(y-1)^2=9x^2[/math] is [math]\sqrt{a}[/math] where [math]a[/math] is an integer, what is the value of [math]a[/math]?

Let [math]A, B,[/math] and [math]C[/math] be three points. [math]B[/math] and [math]C[/math] are on an ellipse. [math]A[/math] is one of the foci and [math]\overline{BC}[/math] passes through the other. If the lengths of the major axis and minor axis of the ellipse are 100 and 80 respectively, what is the perimeter of triangle [math]ABC[/math]?

Consider an ellipse [math]\dfrac{x^2}{9}+\dfrac{y^2}{4}=1[/math]. Let [math]F_1[/math] and [math]F_2[/math] be its foci and [math]P[/math] be a point on the ellipse such that [math]\overline{PF_1}\perp\overline{F_1F_2}[/math]. If the area of triangle [math]PF_1F_2[/math] is [math]a\sqrt{5}[/math] where [math]a[/math] is a rational number, what is the value of [math]a[/math]?

Let [math]a, b[/math], and [math]c[/math] be three positive numbers. Which of the following statement(s) is/are true about the equation [math]ax^2+by^2=c[/math]?

Let [math]P, Q, R[/math] and [math]S[/math] be four points on an ellipse [math]\dfrac{x^2}{9}+\dfrac{y^2}{25}=1[/math]. If [math]PQRS[/math] is a rectangle with center [math](0,0)[/math] and [math]\overline{PQ}[/math] is parallel to the major axis, what is the maximum area of the rectangle?

Consider an ellipse [math]\dfrac{x^2}{25}+\dfrac{y^2}{16}=1[/math]. Let [math]F_1[/math] and [math]F_2[/math] be the foci and [math]P[/math] be a point on the ellipse. If [math]PF_1F_2[/math] is an isosceles triangle and the area of [math]PF_1F_2[/math] is [math]\sqrt{a}[/math] where [math]a[/math] is an integer, what is the minimum of [math]a[/math]?

Consider an ellipse with center [math](3,2)[/math]. The length of the minor axis is [math]6[/math]. Let [math]F_1[/math] and [math]F_2[/math] be the foci and [math]P[/math] be an endpoint of the minor axis. Given that [math]PF_1F_2[/math] is an equilateral triangle, find the equation of the ellipse.