Basics of Conic Section

What type of conic section following quadratic form represent?

$Q\equiv 17{x_1}^2-30x_1{x}_2+17{x_2}^2=128$

Identify the conic section generated when a plane parallel to the axis of cone intersects the solid cone.

Identify the conic section which represents the given equation:

$16x^2+8xy+y^2-74x-78y+212=0$

Identify the conic section represented by the given equation:

$x^2+4xy+y^2-2x+2y-6=0$

Identify the conic section which represents the given equation:

$8x^2+4xy+5y^2-24x-24y=0$

Identify the conic section which represents the given equation:

$16x^2+8xy+y^2-74x-78y+212=0$

STATEMENT-1 : If the distances of a point $\text{'}P\text{'}$ from the line $x+y+2=0$ and point $(3,4)$ are same then the locus of $\text{'}P\text{'}$ will be a parabola.

and  STATEMENT-2 : If the distances of a point $\text{'}P\text{'}$ from a fixed line and a fixed point are same then the locus of $\text{'}P\text{'}$ will be a parabola.

Identify the conic section of the following equation:

$16x^2+8xy+y^2-74x-78y+212=0$

Identify the conic section of the following equation:

$x^2+4xy+y^2-2x+2y-6=0$

Identify the conic section of the following equation:

$x^2+4xy+y^2-2x+2y-6=0$

Identify the conic section of the following equation:

$8x^2+4xy+5y^2-24x-24y=0$

Identify the conic section of the following equation:

$16x^2+8xy+y^2-74x-78y+212=0$

Statement $1:$ If the equation of a conic section is $\frac{x^2}{1-r}-\frac{y^2}{1+r}=1, r>1$ Then it represents a circle
Statement $2:$ A Conic section with eccentricity $e=0$ is a circle.

The equation of the hyperbola of given transverse axis, whose vertex bisects the distance between the centre and the focus, is given by

(Assume the equation of hyperbola as $\frac{x^2}{a^2}-\frac{{\displaystyle y^2}}{{\displaystyle b^2}}=1$)

The locus of the point of contact of tangents drawn from a given point to a system of confocal ellipse is a cubic curve, which passes through the given point and the foci. If the given point be on the major axis, select the correct option in which the cubic reduces to a circle.

Equation $a{x}^2 + 2hxy+b{y}^2+2gx+2fy+c=0$, $(abc+2fgh-a{f}^2- b{g}^2 - c{h}^2\ne 0)$ represents a parabola, if

Cotyledons are also called-

The equation $x^2-2xy+y^2+3x+2$ represents

The second degree equation

${\mathrm{x}}^2+\mathrm{ }4{\mathrm{y}}^2+\mathrm{ }2\mathrm{x}\mathrm{ }+\mathrm{ }16\mathrm{y}\mathrm{ }+13=0$ represent itself as -

Statement$-1$: If the distances of a point $\text{'}P\text{'}$ from the line $x+y+2=0$ and point $(3,4)$ are same then the locus of $\text{'}P\text{'}$ will be a parabola.

Statement$-2$: In $2$-dimensional geometry, if the distances of a point $\text{'}P\text{'}$ from a fixed line and a fixed point are same then the locus of $\text{'}P\text{'}$ will be a parabola.