Math Problems Vault

Problem 559

(1)(2)(3)

Find the necessary and sufficient condition for the line $A x+B y+C = 0$ to:

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(1)

intersect the circle $x^{2}+y^{2}+\alpha x+\beta y+\gamma = 0$ ;

(2)

not intersect this circle;

(3)

be tangent to this circle.

Problem 560

Set up the equation of the tangent to the quadric

a_{11} x^{2}+2 a_{12} x y+a_{22} y^{2}+2 a_{1} x+2 a_{2} y = 0

at the origin.

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Problem 561

Set up the equations of the tangents to the ellipse

\frac{x^{2}}{32}+\frac{y^{2}}{18} = 1

drawn from the point $(12,-3)$ .

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Problem 562

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

Set up the equations of the tangents to the ellipse

\frac{x^{2}}{18}+\frac{y^{2}}{8} = 1

passing through the point:

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(1)

$(-6,0)$ ;

(2)

$(2,7 \sqrt{7})$ ;

(3)

$\left(-4,-\frac{2 \sqrt{2}}{3}\right)$ ;

(4)

$(3,-3)$ .

Problem 563

(1)(2)

Set up the equation of the tangent to the hyperbola $\frac{x^{2}}{9}-\frac{y^{2}}{36} = 1$ :

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(1)

parallel to the line $3 x-y-17 = 0$ ;

(2)

perpendicular to the line $2 x+5 y+11 = 0$ .

Problem 564

Set up the equations of the tangents to the hyperbola

\frac{x^{2}}{4}-y^{2} = 1

passing through the point:

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(1)

$(-2,2)$ ;

(2)

$(1,4)$ ;

(3)

$(4, \sqrt{3})$ ;

(4)

$(4,1)$ ;

(5)

$(8,4)$ ;

(6)

$(0,0)$ .

Problem 565

(1)(2)(3)

Set up the equation of the tangents to the parabola $y^{2} = 16 x$ , passing through the point:

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(1)

$(1,-2)$ ;

(2)

$(1,4)$ ;

(3)

$(1,5)$ .

Problem 566

(1)(2)(3)

Draw tangents to the quadric $2 x^{2}+7 x y+6 y^{2}+3 x + 5 y-26 = 0$ through the points:

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(1)

$(1,1)$ ;

(2)

$(-8,5)$ ;

(3)

$(-7,6)$ .

Problem 567

(1)(2)(3)

Find the conditions necessary and sufficient for it to be possible to draw from the point $\left(x_{0}, y_{0}\right)$ to the hyperbola:

\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}} = 1

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(1)

two tangents;

(2)

one tangent;

(3)

no tangent at all.

Problem 568

Find the necessary and sufficient condition for tangency of the line $A x+B y+C = 0$ and the ellipse

\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}} = 1

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Problem 569

Prove that for any value of $\theta$ the line

\frac{x \cos \theta }{a}+\frac{y \sin \theta }{b} = 1

is tangent to one and the same ellipse. Set up its equation.

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Problem 570

Find the necessary and sufficient condition for tangency of the line $A x+B y+C = 0$ with the hyperbola

\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}} = 1

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Problem 571

Prove that for any value of $\theta$ the line

\frac{x \operatorname {ch} \theta }{a}-\frac{y \operatorname {sh} \theta }{b} = 1

is tangent to one and the same hyperbola, given by its canonical equation. Set up the equation of this hyperbola.

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Problem 572

Find the necessary and sufficient condition for tangency of the line $A x+B y+C = 0$ and the parabola $y^{2} = 2 p x$ .

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Problem 573

(Dual quadric.) Prove that the line $\alpha x+\beta y + 1 = 0$ is tangent to the nondegenerate real quadric (6.1) or is one of its asymptotes if and only if $\alpha$ and $\beta$ satisfy the equation

\left[\begin{smallmatrix} \alpha & \beta & 1 \end{smallmatrix}\right]\left[\begin{smallmatrix} a_{11} & a_{12} & a_{1} \\ a_{12} & a_{22} & a_{2} \\ a_{1} & a_{2} & a \end{smallmatrix}\right]^{-1}\left[\begin{smallmatrix} \alpha \\ \beta \\ 1 \end{smallmatrix}\right] = 0

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Problem 574

Let $\Gamma$ be a second-order curve for which $I_{2} \neq 0$ or $I_{3} \neq 0$ . Prove that the line $A x+B y+C = 0$ is tangent to the curve $\Gamma$ , passes through its singular point, or is an asymptote if and only if the following equality holds:

\left|\begin{smallmatrix} a_{11} & a_{12} & a_{1} & A \\ a_{12} & a_{22} & a_{2} & B \\ a_{1} & a_{2} & a & C \\ A & B & C & 0 \end{smallmatrix}\right| = 0

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Problem 575

Prove that for any value of $k$ the line $y = k x+1/k$ is tangent to one and the same parabola, given by its canonical equation. Find the equation of this parabola.

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Problem 576

Set up the equation of the common tangents to the ellipses

\frac{x^{2}}{45}+\frac{y^{2}}{9} = 1 \quad \text{ and } \quad \frac{x^{2}}{9}+\frac{y^{2}}{18} = 1

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Problem 577

An ellipse having foci at the points $F_{1}(-3,0)$ and $F_{2}(3,0)$ is tangent to the line $x+y-5 = 0$ . Set up the equation of the ellipse.

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Problem 578

Set up the equation of a second-order curve that is tangent to the lines $x = 0, y = 0$ at the points $P(0,3), Q(4,0)$ and to the line $x+y-1 = 0$ . Find the point of tangency with the last line.

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Problem 579

Write the equation of a hyperbola having as asymptotes the lines $x-1 = 0,2 x-y+1 = 0$ and tangent to the line $4 x+y+5 = 0$ .

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Problem 580

Set up the equation of a parabola tangent to the axes $O x$ and $O y$ at the points $P(3,0)$ and $Q(0,5)$ , respectively.

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Problem 581

Set up the equation of a second-order curve tangent to the sides of a triangle $P Q R$ and having its center at the point $S(2,1)$ , if the coordinates of the vertices $P(0,0), Q(5,0), R(0,4)$ are known.

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Problem 582

Prove that the product of the distances from the foci of an ellipse to any tangent to it equals the square of the minor semi-axis.

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Problem 583

Prove that the product of the distances from the foci of a hyperbola to any tangent to it equals the square of the imaginary semi-axis.

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Problem 584

The foci of a hyperbola, $F_{1}(4,2), F_{2}(-1,-10)$ , and the equation of a tangent line, $3 x+4 y-5 = 0$ , are given. Find the semi-axes of the hyperbola.

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Problem 585

Find the locus of the points of intersection of mutually perpendicular tangents to a given ellipse with semi-axes $a, b$ .

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Problem 586

An ellipse with semi-axes $a$ and $b$ , where $a < b$ , moving in the plane remains tangent to the coordinate axes while staying in the first quadrant. What curve does the center of the ellipse trace?

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Problem 587

Find the locus of the vertices of right angles whose sides are tangent to a given hyperbola with semi-axes $a$ and $b$ .

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Problem 588

Find the locus of the feet of the perpendiculars dropped from the focus of a parabola onto the tangents to it.

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