Basics of Conic Section

Which type of conic section is formed when a plane cuts the nappe of the cone and makes an angle $\theta$ with nappe where $\alpha <\theta <90°$? Determine the eccentricity of $\frac{x^2}{16}+\frac{y^2}{25}=1$.$$

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

Find the equation of a conic whose focus is at $\left(1,0\right)$, its eccentricity $\left(e\right)=\frac{1}{\sqrt{2}}$, and directrix is $12x+5y+1=0$.

Identify the conic section which represents the given equation:

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

Identify the path traced by a point $P\left(x,y\right)$ on the curve whose equation is $5\left(x^2+y^2\right)={\left(3x+4y-1\right)}^2$ by using the focus-directrix property of the conic.

Find the equation of a conic whose focus is at $\left(1,0\right)$, its eccentricity $\left(e\right)=\frac{1}{\sqrt{2}}$, and directrix is $12x+5y+1=0$.

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:

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

Identify the path traced by a point $P\left(x,y\right)$ on the curve whose equation is $5\left(x^2+y^2\right)={\left(3x+4y-1\right)}^2$ by using the focus-directrix property of the conic.

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.

Define conic section?

Identify the conic section of the following equation:

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

Identify the conic section of the following equation:

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

Find the equation of the hyperbola with directrix $x+\sqrt{2}y=1$ focus at $\left(1,2\right)$, and eccentricity $2$.

Identify the conic section of the following equation:

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

Find the equation of the hyperbola with directrix $2x+y=1$ focus at $\left(0,0\right)$, and eccentricity $\sqrt{3}$.

Find the equation of the hyperbola with directrix $x+2y=1$ focus at $\left(1,2\right)$, and eccentricity $2$.

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:

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