Name: Condition for Concurrency of Three Lines
Uploaded: 2019-07-19
Duration: 10 min 26 s
Description: Well, this video beautifully explains the conditions for concurrency of three lines with some interesting examples. Come, let's learn and watch the video...!!

Question

If the equation

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

, represents a pair of straight lines, prove that the third pair of straight lines (excluding

x y = 0

) passing through the points where these meet the axes is

a x^{2} - 2 h x y + b y^{2} + 2 g x + 2 f y + c + \frac{4 f g}{c} \cdot x y = 0

.

Accepted Answer

The given pair of lines is $a x^{2} + 2 h x y + b y^{2} + 2 g x + 2 f y + c = 0$ and the equation of the axes is $x y = 0$ .

Using family of lines the pair of equation passing through the point of intersection of the lines $a x^{2} + 2 h x y + b y^{2} + 2 g x + 2 f y + c = 0$ and $x y = 0$ is

$a x^{2} + 2 h x y + b y^{2} + 2 g x + 2 f y + c + λ (x y) = 0 \dots (i)$

$\Rightarrow a x^{2} + x y (2 h + λ) + b y^{2} + 2 g x + 2 f y + c = 0$

This equation is of the form $A x^{2} + 2 H x y + B y^{2} + 2 G x + 2 F y + C = 0$ , which represents a pair of straight lines, if

$A B C + 2 F G H - A F^{2} - B G^{2} - C H^{2} = 0$

$\Rightarrow a b c + 2 f g (\frac{2 h + λ}{2}) - a f^{2} - b g^{2} - c {(\frac{2 h + λ}{2})}^{2} = 0$

$\Rightarrow a b c + 2 f g h + f g λ - a f^{2} - b g^{2} - c (\frac{4 h^{2} + λ^{2} + 4 h λ}{4}) = 0$

$\Rightarrow 4 a b c + 8 f g h + 4 f g λ - 4 a f^{2} - 4 b g^{2} - 4 c h^{2} - c λ^{2} - 4 c h λ = 0$

$\Rightarrow 4 (a b c + 2 f g h - a f^{2} - b g^{2} - c h^{2}) + 4 f g λ - c λ^{2} - 4 c h λ = 0$

$\Rightarrow 0 + 4 f g λ - c λ^{2} - 4 c h λ = 0$

$\Rightarrow λ (4 f g - c λ - 4 c h) = 0$

$\Rightarrow λ = 0$ or $4 f g - c λ - 4 c h = 0$

But for $λ = 0$ , we get the same equation as of the given equation, hence $λ = (\frac{4 f g - 4 c h}{c})$ .

Put, the value of $λ$ in the equation $(i)$ , to get the required equation of the lines is $a x^{2} + 2 h x y + b y^{2} + 2 g x + 2 f y + c + (\frac{4 f g - 4 c h}{c}) (x y) = 0$

$\Rightarrow a x^{2} + 2 h x y + b y^{2} + 2 g x + 2 f y + c + \frac{4 f g}{c} \cdot x y - 4 h x y = 0$

$\Rightarrow a x^{2} - 2 h x y + b y^{2} + 2 g x + 2 f y + c + \frac{4 f g}{c} \cdot x y = 0$ , which is the required equation.

Important Questions on Point and Straight Line

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If the equation ax2+2hxy+by2+2gx+2fy+c=0, represents a pair of straight lines, prove that the third pair of straight lines (excluding xy=0) passing through the points where these meet the axes is ax2-2hxy+by2+2gx+2fy+c+4fgc·xy=0.

Important Questions on Point and Straight Line

Learn and Prepare for any exam you want !