Chapter 10: Q 6. (page 676)

Except for degenerate cases, the equation $A x^{2} + B x y + C y^{2} + D x + E y + F = 0$ defines $a (n)__________$ if $B^{2} - 4 A C = 0$ .

Short Answer

Expert verified

Except for degenerate cases, the equation $A x^{2} + B x y + C y^{2} + D x + E y + F = 0$ defines a parabola if $B^{2} - 4 A C = 0$ .

Step by step solution

Step 1. Given information

An equation is $A x^{2} + B x y + C y^{2} + D x + E y + F = 0$ .

Step 2. What does an equation Ax2+Bxy+Cy2+Dx+Ey+F=0 represents?

If $B^{2} - 4 A C = 0$ , then either $B^{2} = 0 or 4 A C = 0,$

Now it's not a case of degenerate axis, so $B = 0$ here, or

$A \neq 0 or C \neq 0$

but not all three simultaneously, so the form of given equation is either

$\begin{array}{rcl} A x^{2} + D x + E y + F & = & 0, A \neq 0 \\ C y^{2} + D x + E y + F & = & 0, C \neq 0 \end{array}$

$\begin{array}{rcl} A (x^{2} + 2 \times \frac{1}{2} \times \frac{D}{A} x) + E y + F & = & 0 \\ A (x^{2} + 2 \times \frac{1}{2} \times \frac{D}{A} x + \frac{D^{2}}{4 A^{2}}) + E y - F & = & 0 \\ A {(x + \frac{D}{2 A})}^{2} & = & - E y + F + \frac{D^{2}}{4 A^{2}} \end{array}$

On dividing both sides by $A$ ,

$\begin{array}{rcl} \frac{A {(x + \frac{D}{2 A})}^{2}}{A} & = & \frac{1}{A} (- E y + F + \frac{D^{2}}{4 A^{2}}) \\ {(x + \frac{D}{2 A})}^{2} & = & - \frac{E}{A} (y - \frac{F}{E} - \frac{D^{2}}{4 E A^{2}}) \end{array}$

$(x - h)^{2} = - 4 a (y - k)$

Similarly,

$C y^{2} + D x + E y + F = 0 C \neq 0,$ denotes a parabola.

Thus except for degenerate cases, the equation $A x^{2} + B x y + C y^{2} + D x + E y + F = 0$ defines a parabola, if $B^{2} - 4 A C = 0$

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Except for degenerate cases, the equation $A x^{2} + B x y + C y^{2} + D x + E y + F = 0$ defines $a (n)__________$ if $B^{2} - 4 A C = 0$ .

Short Answer

Step by step solution

Step 1. Given information

Step 2. What does an equation Ax2+Bxy+Cy2+Dx+Ey+F=0 represents?

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Except for degenerate cases, the equation Ax2+Bxy+Cy2+Dx+Ey+F=0defines an__________ if B2-4AC=0.

Short Answer

Step by step solution

Step 1. Given information

Step 2. What does an equation Ax2+Bxy+Cy2+Dx+Ey+F=0 represents?

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Probability and Statistics

Mechanics Maths

Discrete Mathematics

Geometry

Pure Maths

Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.

Except for degenerate cases, the equation $A x^{2} + B x y + C y^{2} + D x + E y + F = 0$ defines $a (n)__________$ if $B^{2} - 4 A C = 0$ .