Question

equation of cylinder whose generators intersect the conic ax2+by2+2hxy+2gx+2fy+c=0,z=0​

Answer 1

Answer:

A cylinder is a surface generated by a straight line which is parallel to a fixed line and intersects a given curve or touches a given surface. The fixed line is called the axis and the given curve is called the guiding curve of the cylinder.Any line on the surface of a cylinder is called its generator.

To find the equation of a cylinder whose generators intersect the conic ax2 + 2hxy + by2 + 2gx + 2fy + c = 0, z = 0 ...(1) and are parallel to the line x/l = y/m = z/n. ...(2)

Suppose P (x1 , y1 , z1 ) be any point on the cylinder. The equation of the generator through the point P and parallel to equation (2) are (x- x1)/ l = (y- y1)/ m = (z- z1)/n (3)

The generator (3) meets the plane z = 0 in the point [(x1- lz1)/ n, (y- mz1)/n, 0]

Since the generator (3) meets the given conic (1), at (nx1 – lz1)2 + 2h (nx1 – lz1 ) (ny1 – mz1 ) + b (ny1 – mz1) 2 + 2gn (nx1 – lz1) + 2fn (ny1 – mz1) + cn2 = 0

Thus the locus of (x1 , y1 , z1) is a (nx – lz) 2 + 2h (nx – lz) (ny – mz) + b (ny – mz) 2 + 2gn (nx – lz) + 2fn (ny – mz) + cn2 = 0 This is the required equation of the cylinder.

Answer 2

Concept:

A cylinder is the surface formed when a straight line crosses a fixed planar closed curve while traveling parallel to a fixed straight line.

An axis that is parallel to the main axis of a cylinder and tangential to its surface is said to be the cylinder's generator.

Given:

A cylinder whose generators intersect the conic $ax^2+by^2+2hxy+2gx+2fy+c=0,z=0$ .

Find:

The equation of the cylinder.

Solution:

The equation of the conic which intersects with the generators of the cylinder is given as:

$ax^2+by^2+2hxy+2gx+2fy+c=0,z=0$

The generators of the cylinder are parallel to the lines $\frac{x}{l}=\frac{y}{m}=\frac{z}{n}$ .

Let say, $A(x_1,y_1,z_1)$ is a point of the cylinder.

The equation of the generator which passe through the point $A$ and parallel to $\frac{x}{l}=\frac{y}{m}=\frac{z}{n}$  is:

$\frac{x-x_1}{l}=\frac{y-y_1}{m}=\frac{z-z_1}{n}$

Therefore, this generator intersects the plane $z=0$ at the point $(\frac{nx_1-lz_1}{n},\frac{ny_1-mz_1}{n},0)$.

When the generator meets the conic $ax^2+by^2+2hxy+2gx+2fy+c=0$ the equation will be:

$(nx_1-lz_1)^2 + 2h (nx_1-lz_1 ) (ny_1-mz_1 ) + b (ny_1 - mz_1) ^2 + 2gn (nx_1 - lz_1) + 2fn (ny_1 - mz_1) + cn^2 = 0$

Therefore,

The equation of locus of $(x_1,y_1,z_1)$ is:

$(nx - lz)^2+ 2h (nx - lz) (ny - mz) + b (ny -mz)^2 + 2gn (nx - lz) + 2fn (ny -mz) + cn^2 = 0$

The equation of the cylinder is $(nx - lz)^2+ 2h (nx - lz) (ny - mz) + b (ny -mz)^2 + 2gn (nx - lz) + 2fn (ny -mz) + cn^2 = 0$

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