Question

How do you determine circle, parabola, ellipse, or hyperbola from equation $4{x^2} - 9{y^2} - 16x + 18y - 11 = 0$?

Answer 1

Given an equation. We have to determine whether the equation represents a circle, parabola, ellipse or hyperbola. Then, we will compare the coefficients of ${x^2}$ and ${y^2}$ terms. If they match, then the equation represents the circle. If they don’t match but have both positive coefficients, then the equation represents the equation of ellipse, if the one coefficient is positive and one negative then, it represents the equation of hyperbola. Also, if the equation contains only one squared term, then it represents the parabola.

Formula used:
The standard form of equation of a circle is given by:
${\left( {x - h} \right)^2} + {\left( {y - k} \right)^2} = {r^2}$
The standard form of equation of ellipse is given by:
$\dfrac{{{{\left( {x - h} \right)}^2}}}{{{a^2}}} + \dfrac{{{{\left( {y - k} \right)}^2}}}{{{b^2}}} = 1$
The standard form of equation of hyperbola is given by:
$\dfrac{{{{\left( {x - h} \right)}^2}}}{{{a^2}}} - \dfrac{{{{\left( {y - k} \right)}^2}}}{{{b^2}}} = 1$
The standard form of equation of parabola is given by:
$\left( {y - k} \right) = a{\left( {x - h} \right)^2}$

Complete step by step solution:
We are given an equation, $4{x^2} - 9{y^2} - 16x + 18y - 11 = 0$.
First, we will write the equation in standard form by adding $11$ to both sides of the equation.
$\Rightarrow 4{x^2} - 9{y^2} - 16x + 18y - 11 + 11 = 0 + 11$
$\Rightarrow 4{x^2} - 9{y^2} - 16x + 18y = 11$
Now, we will add $16$ to both sides of equation.
$\Rightarrow 4{x^2} - 9{y^2} - 16x + 18y + 16 = 11 + 16$
Now, we will subtract $9$ from both sides of the equation.
$\Rightarrow 4{x^2} - 9{y^2} - 16x + 18y + 16 - 9 = 11 + 16 - 9$
Now, we will group the terms on the left hand side of the equation.
$\Rightarrow \left( {4{x^2} - 16x + 16} \right) - \left( {9{y^2} - 18y + 9} \right) = 11 + 16 - 9$
We will take out the common terms from each group on the left hand side of the equation.
$\Rightarrow 4\left( {{x^2} - 4x + 4} \right) - 9\left( {{y^2} - 2y + 1} \right) = 18$
Now, apply the algebraic identity ${\left( {a - b} \right)^2} = {a^2} - 2ab + {b^2}$ to the left hand side of the equation.
$\Rightarrow 4{\left( {x - 2} \right)^2} - 9{\left( {y - 1} \right)^2} = 18$
Now, we will divide both sides of the equation by $18$.
$\Rightarrow \dfrac{{4{{\left( {x - 2} \right)}^2}}}{{18}} - \dfrac{{9{{\left( {y - 1} \right)}^2}}}{{18}} = \dfrac{{18}}{{18}}$
$\Rightarrow \dfrac{{{{\left( {x - 2} \right)}^2}}}{{4.5}} - \dfrac{{{{\left( {y - 1} \right)}^2}}}{2} = 1$
Since, the equation contains both ${x^2}$ and ${y^2}$ terms which means the equation represents either ellipse or hyperbola.
Also the coefficients of ${x^2}$ and ${y^2}$ are different but one is negative and one is positive which means the equation represents the hyperbola.
Hence, the equation $4{x^2} - 9{y^2} - 16x + 18y - 11 = 0$ represents the hyperbola.

Note: Please note that while solving the problem we should understand that the coefficient of ${x^2}$ and ${y^2}$ matches, then the equation represents circle or ellipse but the coefficient can be equal or not equal. If they are equal, then it is a circle otherwise it is ellipse. If the one coefficient is positive and one negative then, it represents the equation of hyperbola. Also, if the equation contains only one squared term, then it represents the parabola.