Name: Introduction to Second Degree Equation
Uploaded: 2019-07-19
Duration: 8 min 21 s
Description: In this video, we will learn about the general equation of second degree in two variables, x and y. We will get a basic introduction to second-degree equatio...

Question

Prove that a homogeneous equation of degree two in

x

and

y

i.e.

a x^{2} + 2 h x y + b y^{2} = 0

represents a pair of lines passing through the origin, if

h^{2} - a b \geq 0

.

Accepted Answer

Let $a x^{2} + 2 h x y + b y^{2} = 0 \dots . (i)$ be a homogeneous equation of degree $2$ in $x$ and $y$ .

Case(i):

If $b = 0 (i.e., a \neq 0, h \neq 0)$ , then the equation $(i)$ reduces to $a x^{2} + 2 h x y = 0$ i.e., $x (a x + 2 h y) = 0$ . This represents two lines, $x = 0 and a x + 2 h y = 0$ , both passing through the origin.

Case(ii):

If $a = 0 and b = 0 (i.e., h \neq 0)$ , then the equation $(i)$ reduces to $2 h x y = 0$ , i.e., $x y = 0$ which represents the coordinate axes and they pass through the origin.

Case(iii):

If $b \neq 0$

Multiplying both sides of equation $(i)$ by $b$ , we get

$a b x^{2} + 2 h b x y + b^{2} y^{2} = 0$

$\Rightarrow b^{2} y^{2} + 2 h b x y = - a b x^{2}$

To make L.H.S. a complete square, we add $h^{2} x^{2}$ on both the sides.

$\Rightarrow b^{2} y^{2} + 2 h b x y + h^{2} x^{2} = - a b x^{2} + h^{2} x^{2}$

$\Rightarrow (b y + h x)^{2} = (h^{2} - a b) x^{2}$

$\Rightarrow (b y + h x)^{2} - {[(\sqrt{h^{2} - a b}) x]}^{2} = 0$

$\Rightarrow [(b y + h x) + (\sqrt{h^{2} - a b}) x] \times [(b y + h x) - (\sqrt{h^{2} - a b}) x] = 0$

This is joint equation of two lines

$\Rightarrow (b y + h x) + (\sqrt{h^{2} - a b}) x = 0 and (b y + h x) - (\sqrt{h^{2} - a b}) x = 0$

$\Rightarrow (h + \sqrt{h^{2} - a b}) x + b y = 0 and (h - \sqrt{h^{2} - a b}) x + b y = 0$

These lines pass through the origin.

From the above three cases, we conclude that the equation $a x^{2} + 2 h x y + b y^{2} = 0$ represents a pair of lines passing through the origin, if $h^{2} - a b \geq 0$ .

Prove that a homogeneous equation of degree two in $x$ and $y$ i.e. $a x^{2} + 2 h x y + b y^{2} = 0$ represents a pair of lines passing through the origin, if $h^{2} - a b \geq 0$ .

Important Questions on Pair of Straight Lines

Learn and Prepare for any exam you want !

Prove that a homogeneous equation of degree two in x and y i.e. ax2+2hxy+by2=0 represents a pair of lines passing through the origin, if h2-ab≥0.

Important Questions on Pair of Straight Lines

Learn and Prepare for any exam you want !

Prove that a homogeneous equation of degree two in $x$ and $y$ i.e. $a x^{2} + 2 h x y + b y^{2} = 0$ represents a pair of lines passing through the origin, if $h^{2} - a b \geq 0$ .