Analytic Geometry Answers

Q1) Let S ≡ 4x² −9y²−36 = 0 and S' ≡ y² −4x = 0 be two conics. Under what

conditions on k, will the conic S+kS' = 0 represent:

i) an ellipse?

ii) a hyperbola?

Q2) Find the section of the conicoid x²/2-y²/3= 2z by the plane x−2y+z = 1. What

conic does this section represent? Justify your answer. 

Q3)If 1,1/2,0 are direction ratios of a line, then the line makes an angle of 90◦ with the

x-axis, an angle of 60◦ with the y-axis, and is parallel to the z-axis.

Q4)If a cone has three mutually perpendicular generators then its reciprocal cone has

three mutually perpendicular tangent planes.

Let L₁ be the line in R³ with equation (x,y,z)=(1,0,2) + t(−1,3,4); t∈R and let L₂ be the line that is parallel to L1 and contains the point (1,−1,3). Let V be the plane that contains both the lines L₁ and L₂.

(a)Find two vectors that are both parallel to the plane V but are not parallel to one another.

(b) FindavectorthatisperpendiculartotheplaneV(c) Find an equation for the plane V.(d) Find an equation for the line L₃ that is perpendicular to the plane V and contains the point (1,−1,4).

Peter bought a new graduated cylinder for his chemistry class. It holds 490 mL of liquid. If the cylinder has a radius of 3.5 cm, then how tall is the cylinder?

Use Numerical Method to find Limit of function    Find  Where n is your arid number for example if your arid number is 19-arid-12345 then choose n=12345. Choose at least 4 most nearest values of n for both (Don’t choose far values from n marks will be deducted in that case). Construct neat table and also perform all calculations. And check whether the limit exist or not? If yes then what’s the value of limit. 

The slope of the line which passes through the point (8,14) and (0,4) is

. (a) The equation ax + by = 0 represents a line through the origin in R2. Show that the vector n1 = (a, b) formed from the coefficients of the equation is orthogonal to the line, that is, orthogonal to every vector along the line. (b) The equation ax + by + cz = 0 represents a plane through the origin in R3. Show that the vector n2 = (a, b, c) formed from the coefficients of the equation is orthogonal to the plane, that is, orthogonal to every vector that lies in the plane.

1. (Sections 2.3, 2.10, 2.11, 2.12) Let L1 be the line in R 3 with equation (x, y, z) = (1, 0, 2) + t(−1, 3, 4) ; t ∈ R and let L2 be the line that is parallel to L1 and contains the point (1, −1, 3). Let V be the plane that contains both the lines L1 and L2. (a) Find two vectors that are both parallel to the plane V but are not parallel to one another. (2) (b) Find a vector that is perpendicular to the plane V . (2) (c) Find an equation for the plane V . (2) (d) Find an equation for the line L3 that is perpendicular to the plane V and contains the point (1, −1, 4) . (2) Hint: Find a parametric equation for L3. Don’t try to find a Cartesian equation for L3. (Study Remarks 2.12.2.)

find the shape of a c curved mirror such that lightfrom the source at the origin will be reflected in a beam of rays parallel to x-axis.

If 1, 1/2, 0 are direction ratios of a line, then the line makes an angle of 90° with the x-axis, an angle of 60° with the y-axis, and is parallel to the z-axis. State whether true or false, also give reason for your answer.

Ferris wheel cars are at a position of A (9,33) and B (27,-15). One of its axle is

represented by 3y+2x-7= 0 where it passes through the center of the wheel.

Draw the suitable graph of the said situation.