Question

The equation $2{y^2} - xy - {x^2} + 6x - 8$ represents
A. A pair of straight lines
B. A circle
C. An ellipse
D. A parabola

Answer 1

Hint: In the given problem, we need to determine which conic the given equation represents. The general equation of a conic is $a{x^2} + 2hxy + b{y^2} + 2gx + 2fy + c = 0$. We will compare both the equations and find the value of variables. After this, we will apply conditions for each conic one by one to get our required answer.

Formula Used:
General equation of a conic is $a{x^2} + 2hxy + b{y^2} + 2gx + 2fy + c = 0$

Complete step by step solution:
Given that, $2{y^2} - xy - {x^2} + 6x - 8$
Now, compare the above written equation with the general equation of a conic, $a{x^2} + 2hxy + b{y^2} + 2gx + 2fy + c = 0$.
Thus, on comparing, we get $a = - 1$, $h = - \dfrac{1}{2}$, $b = 2$, $g = 3$, $f = 0$ and $c = - 8$
For straight line: $abc + 2fgh - a{f^2} - b{g^2} - c{h^2} = 0$
On substituting the values, we get
$ = - 1 \times 2 \times - 8 + 2 \times 0 \times 3 \times - \dfrac{1}{2} - \left( { - 1} \right) \times {0^2} - 2 \times {3^2} - \left( { - 8} \right){\left( { - \dfrac{1}{2}} \right)^2}$
On simplification of the above written expression, we get
$ = 16 + 0 - 18 + 2$
$ = 0$
Thus, the condition satisfies. The given equation represents a pair of straight lines.
For circle: $abc + 2fgh - a{f^2} - b{g^2} - c{h^2} \ne 0$, $a = b$ and $h = 0$
In the given case, $a \ne b$. Thus, the given equation is not a circle.
For an ellipse: $abc + 2fgh - a{f^2} - b{g^2} - c{h^2} \ne 0$ and ${h^2} < ab$
$ \Rightarrow {\left( { - \dfrac{1}{2}} \right)^2} < - 1 \times 2$
$ \Rightarrow \dfrac{1}{4} < - 2$, which is wrong.
Thus, the given equation is not of a parabola.
For parabola: $abc + 2fgh - a{f^2} - b{g^2} - c{h^2} \ne 0$ and ${h^2} = ab$
$ \Rightarrow {\left( { - \dfrac{1}{2}} \right)^2} \ne - 1 \times 2$
Thus, the given equation is not of a parabola.

Option ‘A’ is correct

Note: To solve these types of questions, one must remember the general equation of a conic. Also, remember all the conditions for different conics.
An equation represents:
A pair of straight line if $abc + 2fgh - a{f^2} - b{g^2} - c{h^2} = 0$
A circle if $a{x^2} + 2hxy + b{y^2} + 2gx + 2fy + c \ne 0$, $a = b$ and $h = 0$
A parabola if $a{x^2} + 2hxy + b{y^2} + 2gx + 2fy + c \ne 0$ and ${h^2} = ab$
An ellipse if $a{x^2} + 2hxy + b{y^2} + 2gx + 2fy + c \ne 0$ and ${h^2} < ab$
A hyperbola if $a{x^2} + 2hxy + b{y^2} + 2gx + 2fy + c \ne 0$ and ${h^2} > ab$
A rectangular hyperbola if $a{x^2} + 2hxy + b{y^2} + 2gx + 2fy + c \ne 0$, ${h^2} > ab$ and $a + b = 0$.