Question

How do we identify the given Equation is hyperbola , parabola , circle , ellipse etc Without finding h², ab , delta What is the form of those Equations Explain it clearly​

Answer 1

Ellipse

• The only difference between the circle and the ellipse is that in an ellipse, there are two radius measures, one horizontally along the x-axis, the other vertically along the y-axis. Clearly, for a circle both these have the same value. By convention, the y radius is usually called b and the x radius is called a.

Hyperbola

• How To: Given the equation of a hyperbola in standard form, locate its vertices and foci.

1. Solve for a using the equation a=√a2 a = a 2 .
2. Solve for c using the equation c=√a2+b2 c = a 2 + b 2 .

parabola

sorry i don't know

Answer 2

$\large\underline{\sf{Solution-}}$

The general equation of conic is given by

$\sf \: {ax}^{2} + 2hxy + {by}^{2} + 2gx + 2fy + c = 0$

1. Its represents a Circle, if

$\sf \: coefficient \: of \: {x}^{2} = coefficient \: of \: {y}^{2} \: i.e. \: a = b$

and

$\sf \: h = 0$

2. Its represents an ellipse if

$\sf \: coefficient\:of\:{x}^{2} \: and \: {y}^{2} \: should \: be \: of \: same \: sign \: i.e.$

$\sf \: a, \: b > 0 \: and \: a \ne \: b$

3. It represents Hyperbola if

$\sf \: coefficient\:of\:{x}^{2} \: and \: {y}^{2} \: should \: be \: of \:opposite \: sign \: i.e.$

$\sf \: a > 0 \:and \: b < 0 \: \: \red{or} \: a < 0 \: and \: b > 0$

4. It represents Parabola if

$\sf \: either \: the \: coefficient \: of \: {x}^{2} \: or \: {y}^{2} \: is \: 0 \: i.e.$

$\sf \: a \: = \: 0 \: \: \: \red{or} \: \: \: b \: = \: 0$

The general equations in standard form is given as

1. For circle

$\rm :\longmapsto\: {(x - h)}^{2} + {(y - k)}^{2} = {r}^{2}$

where,

$\red{ \sf \: (h,k) \: are \: centre \: and \: r \: is \: radius}$

2. For ellipse

$\rm :\longmapsto\:\dfrac{ {(x - h)}^{2} }{ {a}^{2} } + \dfrac{ {(y - k)}^{2} }{ {b}^{2} } = 1 \: \{a \: \ne \: b \}$

3. For Hyperbola

$\rm :\longmapsto\:\dfrac{ {(x - h)}^{2} }{ {a}^{2} } - \dfrac{ {(y - k)}^{2} }{ {b}^{2} } = 1 \:$

or

$\rm :\longmapsto\: - \: \dfrac{ {(x - h)}^{2} }{ {a}^{2} } + \dfrac{ {(y - k)}^{2} }{ {b}^{2} } = 1 \:$

4. For Parabola with horizontal axis

$\rm :\longmapsto\: {(y - k)}^{2} = 4a(x - h)$

or

Parabola with vertical axis

$\rm :\longmapsto\: {(x - h)}^{2} = 4a(y - k)$

For example :-

$\sf \: 1. \: {x}^{2} + {2y}^{2} = 1 \: represents \: ellipse \: as \:$

$\sf \: coefficient \: of \: {x}^{2} \: and \: {y}^{2} \: are \: not \: same.$

$\sf \: 2. \: {x}^{2} - {2y}^{2} = 1 \: represents \: hyperbola \: as \:$

$\sf \: coefficient \: of \: {x}^{2} \: and \: {y}^{2} \: are \: of \: opposite \: sign.$