Name: Sum and Product of the Slopes of Pair of Straight Lines
Uploaded: 2019-07-19
Duration: 5 min 24 s
Description: A combined equation of a pair of straight lines is obtained by multiplying the equations of two straight lines. Let's learn about their sum and product of sl...

Question

Show that the lines

x^{2} - 4 x y + y^{2} = 0

and

x + y = \sqrt{6}

form an equilateral triangle. Find its area and perimeter.

Accepted Answer

Equation of the pair of lines is $x^{2} - 4 x y + y^{2} = 0$ -----(i)

The point of intersection of the above lines is $O (0, 0)$

Let $θ$ be the angle between the lines then

$∴ \tan θ = \frac{2 \sqrt{h^{2} - a b}}{a + b}$

$∴ \tan θ = \frac{2 \sqrt{{(- 2)}^{2} - 1 \times 1}}{1 + 1}$ $= \frac{2 \sqrt{4 - 1}}{2}$

$∴ \tan θ = \sqrt{3}$

$∴ θ = \tan^{- 1} \sqrt{3}$

$∴ θ = 60^{°}$

Let $O A$ or $O B$ has slope m,

$∴$ Its equation is $y = m x$

Now the slope of $x + y = \sqrt{6}$ is $- 1$

Let us find the lines making $60^{\circ}$ angle with the above line.

$∴ \tan 60^{\circ} = |\frac{m - (- 1)}{1 + m (- 1)}|$

$∴ \sqrt{3} = |\frac{m + 1}{1 - m}|$

Squaring both sides, we get

$∴ 3 = \frac{(m + 1)^{2}}{(1 - m)^{2}}$

$∴ 3 (1 - 2 m + m^{2}) = m^{2} + 2 m + 1$

$∴ 3 - 6 m + 3 m^{2} = m^{2} + 2 m + 1$

$∴ 2 m^{2} - 8 m + 2 = 0$

$∴ m^{2} - 4 m + 1 = 0$

$∴ (\frac{y}{x}) - 4 (\frac{y}{x}) + 1 = 0$

$∴ y^{2} - 4 x y + x^{2} = 0$

$∴ x^{2} - 4 x y + y^{2} = 0$ is the joint equation of two lines through the origin each making an angle of $60^{\circ}$ with $x + y = \sqrt{6}$

$∴ x^{2} - 4 x y + y^{2} = 0$ and $x + y = \sqrt{6}$ form a triangle $O A B$ which is equilateral

Let $P =$ the length of the perpendicular from $O$ to line $x + y = \sqrt{6}$ is

$∴ \frac{|c|}{a^{2 +} b^{2}}$

$∴ |\frac{\sqrt{6}}{\sqrt{1 + 1}}| = \sqrt{3}$

$∴$ Area of $ΔO A B = \frac{(P)^{2}}{\sqrt{3}} = \frac{(\sqrt{3})^{2}}{\sqrt{3}}$

$= \frac{3}{\sqrt{3}}$

$= \sqrt{3} sq units$

In right angled-triangle $O A P$

$∴ \sin 60^{\circ} = \frac{O P}{O A}$

$∴ \frac{\sqrt{3}}{2} = \frac{\sqrt{3}}{O A}$

$∴ O A = 2$

$∴$ Length of each side of an equilateral triangle $O A B = 2 units$

$∴$ Perimeter of $ΔOAB = 3 \times$ length of each side

$= 3 \times 2 = 6 units$

Show that the lines $x^{2} - 4 x y + y^{2} = 0$ and $x + y = \sqrt{6}$ form an equilateral triangle. Find its area and perimeter.

Important Questions on Pair of Straight Lines

Learn and Prepare for any exam you want !

Show that the lines x2-4xy+y2=0 and x+y=6 form an equilateral triangle. Find its area and perimeter.

Important Questions on Pair of Straight Lines

Learn and Prepare for any exam you want !

Show that the lines $x^{2} - 4 x y + y^{2} = 0$ and $x + y = \sqrt{6}$ form an equilateral triangle. Find its area and perimeter.