Math Problems Vault

Problem 271

Let $(x,y)$ be the midpoint of a chord of the curve $f(x,y,1)=0$ , and let $(x+\xi t, y+\eta t)$ and $(x-\xi t, y-\eta t)$ be the endpoints of the chord. Prove that

\left(\frac{\partial f}{\partial x}\right)_{1}\xi + \left(\frac{\partial f}{\partial y}\right)_{1}\eta = 0.

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Remark.

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The polynomial $Ax^{2}+Bxy+Cy^{2}+Dx+Ey+F$ , homogenized by introducing powers of the variable $z$ , becomes the ternary quadratic form $f = f(x,y,z) = Ax^{2}+Bxy+Cy^{2}+Dxz+Eyz+Fz^{2}$ . Its partial derivatives with respect to $x$ , $y$ and $z$ are given by: $\frac{\partial f}{\partial x} = 2Ax+By+Dz$ ; $\frac{\partial f}{\partial y} = Bx+2Cy+Ez$ ; $\frac{\partial f}{\partial z} = Dx+Ey+2Fz$ . Their values at $z=1$ are denoted by $\left(\frac{\partial f}{\partial x}\right)_{1}$ , $\left(\frac{\partial f}{\partial y}\right)_{1}$ and $\left(\frac{\partial f}{\partial z}\right)_{1}$ . For brevity one also writes $f(x,y,1) = Ax^{2}+Bxy+Cy^{2}+Dx+Ey+F$ , $f(x,y,0) = Ax^{2}+Bxy+Cy^{2}$ .

Problem 272

Prove that the midpoints of the chords of the curve $f(x,y,1)=0$ parallel to the line $\frac{x}{\xi } = \frac{y}{\eta }$ lie on the line $\left(\frac{\partial f}{\partial x}\right)_{1}\xi + \left(\frac{\partial f}{\partial y}\right)_{1}\eta = 0$ , or, in full form:

(2Ax+By+D)\xi + (Bx+2Cy+E)\eta = 0.

(The diameter conjugate to the given direction.)

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Problem 273

Show that when the origin is translated to the point $(\xi ,\eta )$ , the equation $f(x,y,1)=0$ passes into the equation

f(x_{1},y_{1},0) + \frac{\partial f}{\partial \xi }x_{1} + \frac{\partial f}{\partial \eta }y_{1} + f(\xi ,\eta ,1) = 0.

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Problem 274

If $4AC-B^{2} \neq 0$ , all the diameters pass through the point $(\xi ,\eta )$ where the lines $\left(\frac{\partial f}{\partial x}\right)_{1}=0$ , $\left(\frac{\partial f}{\partial y}\right)_{1}=0$ intersect. Show that when the origin is translated to the point chosen in this way, the equation $f(x,y,1)=0$ passes into the following one:

Ax_{1}^{2}+Bx_{1}y_{1}+Cy_{1}^{2} + \frac{1}{2}(D\xi +E\eta +2F) = 0.

Here $D\xi +E\eta +2F$ is the value of $\left(\frac{\partial f}{\partial z}\right)_{1}$ at $x=\xi$ , $y=\eta$ .

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Problem 275

Translate the origin to the center of the curve $x^{2}+3xy-2y^{2}-7x-2y+16=0$ and set up the new equation of the curve.

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Problem 276

Simplify the equation of the curve

x^{2}+3xy-2y^{2}-7x-2y+16=0,

by translating the origin to the center.

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Problem 277

Simplify the equation of the curve

x^{2}-2xy+y^{2}-4x+4y-3=0,

by translating the origin to that center of it whose abscissa equals one.

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Problem 278

Show that the line $7x+y+6=0$ passes through the centers of the curves

3x^{2}-7xy-6y^{2}+3x-9y+5=0

and

3x^{2}-5xy+6y^{2}+11x-17y+13=0.

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Problem 279

Find the axes of symmetry of the ellipse

x^{2}+xy-y^{2}-x+y-1=0.

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Problem 280

Find the foci (with respect to the old coordinate axes) of the hyperbola $x^{2}-6xy+y^{2}-2x-2y+5=0$ .

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Problem 281

Determine the eccentricity of the curve

x^{2}-4xy-2y^{2}+10x+4y=0.

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Problem 282

Simplify the equation of the curve

x^{2}-xy+y^{2}-5x+y-2=0,

by translating the origin to the center, and then rotating the axes through the appropriate angle.

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Problem 283

The same, for the curve $x^{2}+6xy+y^{2}+8x+24y+39=0$ .

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Problem 284

Simplify the equation of the parabola

x^{2}-2xy+y^{2}-2x-2y+1=0,

by rotating the axes so that the new equation contains no $x_{1}y_{1}$ , and then translating the origin so that only two terms remain in the final equation.

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Problem 285

Prove that when $4AC-B^{2}=0$ , $A>0$ (so that one may set $A=a^{2}$ ), the identity holds

Ax^{2}+Bxy+Cy^{2}+Dx+Ey+F = (ax+by+h)^{2}+\alpha x+\beta y+\gamma ,

where $\alpha = D-2ah$ , $\beta = E-2bh$ , $\gamma = F-h^{2}$ .

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Problem 286

Prove that in the identity of the preceding problem, the number $h$ can be chosen so that the lines $ax+by+h=0$ and $\alpha x+\beta y+\gamma =0$ are mutually perpendicular, provided only that the numbers $D$ and $E$ are not proportional to the numbers $a$ and $b$ .

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Problem 287

Simplify the equation of the curve

x^{2}+2xy+y^{2}-9x+4=0

using the identity:

x^{2}+2xy+y^{2}-9x+4 = (x+y+h)^{2}+\alpha x+\beta y+\gamma ,

choosing $h$ so that the lines $\alpha x+\beta y+\gamma =0$ and $x+y+h=0$ are mutually perpendicular. Here the formulas for the coordinate transformation have the form:

y_{1} = \frac{x+y+h}{\sqrt{2}}, \qquad x_{1} = \frac{\alpha x+\beta y+\gamma }{\sqrt{\alpha ^{2}+\beta ^{2}}}.

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Problem 288

By the same method, simplify the equation of the parabola

16x^{2}+24xy+9y^{2}-170x+310y-465=0.

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Problem 289

Simplify the equation of the parabola

9x^{2}+30xy+25y^{2}+4x-16y+8=0.

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Problem 290

For given $A$ , $B$ and $C$ , and a suitably chosen value of $\lambda$ , the identity holds

Ax^{2}+Bxy+Cy^{2}-\lambda (x^{2}+y^{2}) = (\alpha x+\beta y)^{2},

where $\alpha$ and $\beta$ are real or imaginary coefficients. Prove that such values of $\lambda$ are the roots of the equation

4\lambda ^{2}-4(A+C)\lambda +4AC-B^{2}=0.

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Problem 291

The equation

Ax^{2}+Bxy+Cy^{2}+Dx+Ey+F = 0

represents a pair of lines, real or imaginary, if and only if the discriminant $\Delta$ equals zero. Using this, prove that under any coordinate transformation the quantity $\Delta$ does not change.

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Remark.

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The discriminant $\Delta$ of the quadratic form in three variables $f(x,y,z) = Ax^{2}+Bxy+Cy^{2}+Dxz+Eyz+Fz^{2}$ is defined as the determinant of the coefficients of $x$ , $y$ and $z$ in the formulas expressing its partial derivatives: $\Delta = \begin{vmatrix} 2A & B & D \\ B & 2C & E \\ D & E & 2F \end{vmatrix}$ . Its minor $\delta = \begin{vmatrix} 2A & B \\ B & 2C \end{vmatrix} = 4AC-B^{2}$ is the discriminant of the quadratic form in two variables $Ax^{2}+Bxy+Cy^{2}$ .

Problem 292

Using the invariance of the quantities $A+C$ , $4AC-B^{2}$ and the discriminant $\Delta$ , prove that when $4AC-B^{2} \neq 0$ , the equation

Ax^{2}+Bxy+Cy^{2}+Dx+Ey+F=0

reduces, after simplification, to the form

\lambda _{1}x_{1}^{2}+\lambda _{2}x_{2}^{2} = \frac{\Delta }{B^{2}-4AC}.

Here $\lambda_{1}$ and $\lambda_{2}$ are the roots of the equation $\lambda^{2}-2(A+C)\lambda +4AC-B^{2}=0$ .

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Problem 293

Prove that the equation $Ax^{2}+Bxy+Cy^{2}+Dx+Ey+F=0$ , when $AC-D^{2}=0$ and $\Delta \neq 0$ , can be reduced to the form

(A+C)x_{1}^{2} = \pm \sqrt{\frac{-\Delta }{2(A+C)}}y_{1}.

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Problem 294

Reduce the equation of the following line to its simplest form: $5x^{2}+4xy+8y^{2}-32x-56y+80=0$ .

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Problem 295

Reduce the equation of the following line to its simplest form: $5x^{2}+8xy+5y^{2}-18x-18y+9=0$ .

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Problem 296

Reduce the equation of the following line to its simplest form: $5x^{2}+6xy+5y^{2}-16x-16y+16=0$ .

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Problem 297

Reduce the equation of the following line to its simplest form: $5x^{2}-4xy+5y^{2}+4x-10y=319$ .

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Problem 298

Reduce the equation of the following line to its simplest form: $6xy+8y^{2}-12x-26y+11=0$ .

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Problem 299

Reduce the equation of the following line to its simplest form: $7x^{2}+16xy-23y^{2}-14x-16y-218=0$ .

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Problem 300

Reduce the equation of the following line to its simplest form: $7x^{2}+24xy+38x+24y+175=0$ .

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Problem 301

Reduce the equation of the following line to its simplest form: $x^{2}-8xy+7y^{2}+6x-6y+9=0$ .

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Problem 302

Reduce the equation of the following line to its simplest form: $9x^{2}+24xy+16y^{2}-40x+30y=0$ .

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Problem 303

Reduce the equation of the following line to its simplest form: $2x^{2}-72xy+23y^{2}+68x+26y+28=0$ .

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