Solution below

Question

asked 2021-08-10

Show that the graph of an equation of the form

\(\displaystyle{A}{x}^{{2}}+{C}{y}^{{2}}+{D}{x}+{E}{y}+{F}={0},{A}\ne{0},{C}\ne{0}\)

where A and C are of opposite sign,

(a) is a hyperbola if \(\displaystyle{\frac{{{D}^{{2}}}}{{{4}{A}}}}+{\frac{{{E}^{{2}}}}{{{4}{C}}}}-{F}\ne{0}\)

(b) is two intersecting lines if \(\displaystyle{\frac{{{D}^{{2}}}}{{{4}{A}}}}+{\frac{{{E}^{{2}}}}{{{4}{C}}}}-{F}={0}\)

\(\displaystyle{A}{x}^{{2}}+{C}{y}^{{2}}+{D}{x}+{E}{y}+{F}={0},{A}\ne{0},{C}\ne{0}\)

where A and C are of opposite sign,

(a) is a hyperbola if \(\displaystyle{\frac{{{D}^{{2}}}}{{{4}{A}}}}+{\frac{{{E}^{{2}}}}{{{4}{C}}}}-{F}\ne{0}\)

(b) is two intersecting lines if \(\displaystyle{\frac{{{D}^{{2}}}}{{{4}{A}}}}+{\frac{{{E}^{{2}}}}{{{4}{C}}}}-{F}={0}\)

asked 2021-08-13

Find an equation of the conic described.Graph the equation.

Ellipse; center at (0,0); focus at (0,3); vertex at (0, 5)

Ellipse; center at (0,0); focus at (0,3); vertex at (0, 5)

asked 2021-07-04

asked 2021-06-10

Determine whether the given set S is a subspace of the vector space V.

A. V=\(P_5\), and S is the subset of \(P_5\) consisting of those polynomials satisfying p(1)>p(0).

B. \(V=R_3\), and S is the set of vectors \((x_1,x_2,x_3)\) in V satisfying \(x_1-6x_2+x_3=5\).

C. \(V=R^n\), and S is the set of solutions to the homogeneous linear system Ax=0 where A is a fixed m×n matrix.

D. V=\(C^2(I)\), and S is the subset of V consisting of those functions satisfying the differential equation y″−4y′+3y=0.

E. V is the vector space of all real-valued functions defined on the interval [a,b], and S is the subset of V consisting of those functions satisfying f(a)=5.

F. V=\(P_n\), and S is the subset of \(P_n\) consisting of those polynomials satisfying p(0)=0.

G. \(V=M_n(R)\), and S is the subset of all symmetric matrices

A. V=\(P_5\), and S is the subset of \(P_5\) consisting of those polynomials satisfying p(1)>p(0).

B. \(V=R_3\), and S is the set of vectors \((x_1,x_2,x_3)\) in V satisfying \(x_1-6x_2+x_3=5\).

C. \(V=R^n\), and S is the set of solutions to the homogeneous linear system Ax=0 where A is a fixed m×n matrix.

D. V=\(C^2(I)\), and S is the subset of V consisting of those functions satisfying the differential equation y″−4y′+3y=0.

E. V is the vector space of all real-valued functions defined on the interval [a,b], and S is the subset of V consisting of those functions satisfying f(a)=5.

F. V=\(P_n\), and S is the subset of \(P_n\) consisting of those polynomials satisfying p(0)=0.

G. \(V=M_n(R)\), and S is the subset of all symmetric matrices