Math Problems Vault

Problem 7.21

A point $A$ lies outside an ellipse with foci $F_{1}, F_{2}$ , the segments $AF_{1}, AF_{2}$ meet the ellipse at points $B, D$ respectively, and $C$ is the point of intersection of the segments $F_{1} D, F_{2} B$ . Prove that a circle can be inscribed in the quadrilateral $ABCD$ .

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Problem 7.22

(1)(2)(3)(4)

Find the lengths of the semi-axes, the eccentricity, the coordinates of the foci, and find the equations of the directrices of the ellipse:

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(1)

$\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1, a > b > 0$

(2)

$\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1, b > a > 0$

(3)

$9x^{2} + 25y^{2} = 225$

(4)

$4x^{2} + y^{2} = 1$

Problem 7.23

(1)(2)(3)

Given the ellipse $25x^{2} + 144y^{2} = 1$ . Determine whether the point $A$ lies on the ellipse, inside it, or outside it:

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(1)

$A(1, 1 / 6)$

(2)

$A(1 / 13, 1 / 13)$

(3)

$A(1 / 6, - 1 / 24)$

Problem 7.24

Compute the length of the focal chord of the ellipse $\frac{x^{2}}{9} + \frac{y^{2}}{4} = 1$ , perpendicular to the major axis.

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Problem 7.25

In a given coordinate system an ellipse has a canonical equation. Find this equation, if:

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(1)

the distance between the vertices lying on the major axis equals 16, and the distance between the foci equals 10

(2)

the chord joining two vertices of the ellipse has length 5 and is inclined to its major axis at an angle $\arcsin (3 / 5)$

(3)

the foci of the ellipse are the points $(\pm 1, 0)$ , and the point $(\sqrt{3}, \sqrt{3} / 2)$ belongs to the ellipse

(4)

the foci of the ellipse are the points $(\pm 2, 0)$ , and the directrices are the lines $x = \pm 18$

(5)

the distance from a directrix to the nearest vertex equals 4, and to the vertex lying on the axis $Oy$ equals 8

(6)

the triangle with vertices at the foci and at the end of the minor axis is equilateral, and the diameter of the circle passing through the center and two vertices of the ellipse equals 7

(7)

the segment of the axis $Ox$ between the focus $F_{1}$ and the far vertex $A$ of the major axis is bisected by the second focus $F_{2}$ , and the distance from $F_{2}$ to the line passing through $A$ and the vertex of the minor axis equals $1 / \sqrt{17}$

(8)

the directrices of the ellipse are the lines $x = \pm 4$ , and the quadrilateral with vertices at the foci and the ends of the minor axis is a square

(9)

the eccentricity of the ellipse equals $\sqrt{7} / 4$ , and the quadrilateral whose vertices are the vertices of the ellipse is circumscribed about a circle of radius 4.8.

Problem 7.26

Compute the eccentricity of the ellipse, if:

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(1)

the distance between the foci equals the arithmetic mean of the lengths of the axes

(2)

the segment between a focus and the far vertex of the major axis is divided by the second focus in the ratio $2: 1$

(3)

the distance from a focus to the far vertex of the major axis is 1.5 times the distance to the vertex of the minor axis

(4)

the segment between the foci is seen from the end of the minor axis at a right angle

(5)

the major axis is seen from the end of the minor axis at an angle of $120^{\circ }$

(6)

the segment between a focus and the far vertex of the major axis is seen from the end of the minor axis at a right angle

(7)

the sides of a square inscribed in the ellipse pass through the foci of the ellipse.

Problem 7.27

Find the equations of the sides of a square inscribed in the ellipse $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1, (a > b > 0)$ . What fraction of the area bounded by the ellipse is the area of this square?

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Problem 7.28

Find the set of points that are the midpoints of chords of the ellipse $\frac{x^{2}}{25} + \frac{y^{2}}{9} = 1$ parallel to the line $x + 2y = 1$ .

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Problem 7.29

Through the point $A(7 / 2, 7 / 4)$ draw a chord of the ellipse $x^{2} + 4y^{2} = 25$ that is bisected at this point.

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Problem 7.30

Through the point $M(0, 3)$ draw a line meeting the ellipse $x^{2} + 4y^{2} = 20$ at two points $A$ and $B$ such that $\left|MA\right| = 2\left|MB\right|$ .

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Problem 7.31

(1)(2)(3)

On the ellipse $\frac{x^{2}}{4} + y^{2} = 1$ find the points from which the segment joining the foci is seen:

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(1)

at a right angle

(2)

at an angle of $60^{\circ }$

(3)

at the largest angle.

Problem 7.32

(1)(2)

Find the equations of the families of ellipses:

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(1)

with common foci $(\pm c, 0)$

(2)

with common directrices $x = \pm d$ and common center at the origin.

Problem 7.33

(1)(2)(3)

Find the equation of an ellipse, if:

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(1)

the points $F_{1}(5, 1)$ and $F_{2}(- 1, 1)$ are the foci, and the line $x = 31 / 3$ is one of the directrices

(2)

the point $F(- 6, 2)$ is one of the foci, the point $A(2, 2)$ is an endpoint of the major axis, and the eccentricity equals $2 / 3$

(3)

the axes of the ellipse are parallel to the coordinate axes, the points $A(4, 0)$ and $B(0, 4)$ belong to the ellipse, and the point $B$ is at distance $3\sqrt{2}$ from one of the foci and at distance 6 from the corresponding directrix.

Problem 7.34

(1)(2)

Let $O$ be the center of an ellipse, $a, b$ its semi-axes, and $A$ and $B$ points of the ellipse such that the lines $OA$ and $OB$ are mutually perpendicular.

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(1)

Prove that the quantity $\frac{1}{\left|OA\right|^{2}} + \frac{1}{\left|OB\right|^{2}}$ is constant for all possible pairs of points $A$ and $B$ .

(2)

Find the largest and smallest values of the length of the segment $AB$ .