Show that the area of the triangle formed by the lines y-m1 x- y-m2 x and y-c is equal to c 2433-11- where m1- m2 are the roots of the equation x 2-3-2 x-3-1-0

The given lines are as follows: y = m1 x nbsp; nbsp; nbsp; nbsp; nbsp; ... (1) y = m2 x nbsp; nbsp; nbsp; nbsp; ... (2) y = c nbsp; nb ...

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Byju's Answer

Standard XI

Mathematics

Point Slope Form of a Line

Show that the...

Question

Show that the area of the triangle formed by the lines y = m₁ x, y = m₂ x and y = c is equal to $\frac{c^{2}}{4} (\sqrt{33} + \sqrt{11}),$ where m₁, m₂ are the roots of the equation $x^{2} + (\sqrt{3} + 2) x + \sqrt{3} - 1 = 0 .$

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Solution

The given lines are as follows:

y = m₁ x ... (1)

y = m₂ x ... (2)

y = c ... (3)

Solving (1) and (2), we get (0, 0) as their point of intersection.

Solving (1) and (3), we get $(\frac{c}{m_{1}}, c)$ as their point of intersection.

Similarly, solving (2) and (3), we get $(\frac{c}{m_{2}}, c)$ as their point of intersection.

∴ Area of the triangle formed by these lines = $\frac{1}{2} |\begin{matrix} 0 & 0 & 1 \\ \frac{c}{m_{1}} & c & 1 \\ \frac{c}{m_{2}} & c & 1 \end{matrix}| = \frac{1}{2} (\frac{c^{2}}{m_{1}} - \frac{c^{2}}{m_{2}}) = \frac{c^{2}}{2} |\frac{m_{2} - m_{1}}{m_{1} m_{2}}|$

It is given that m₁ and m₂ are the roots of the equation $x^{2} + (\sqrt{3} + 2) x + \sqrt{3} - 1 = 0 .$

$∴ m_{1} + m_{2} = - (\sqrt{3} + 2), m_{1} m_{2} = \sqrt{3} - 1 \Rightarrow m_{2} - m_{1} = \sqrt{{(m_{1} + m_{2})}^{2} - 4 m_{1} m_{2}} \Rightarrow m_{2} - m_{1} = \sqrt{{\{- (\sqrt{3} + 2)\}}^{2} - 4 \sqrt{3} + 4} \Rightarrow m_{2} - m_{1} = \sqrt{7 + 4 \sqrt{3} - 4 \sqrt{3} + 4} = \sqrt{11}$

$∴ Area = \frac{c^{2}}{2} |\frac{\sqrt{11}}{\sqrt{3} - 1}| = \frac{c^{2}}{2} |\frac{(\sqrt{3} + 1) \sqrt{11}}{(\sqrt{3} + 1) (\sqrt{3} - 1)}| = \frac{c^{2}}{2} |\frac{(\sqrt{33} + \sqrt{11})}{2}| = \frac{c^{2}}{4} (\sqrt{33} + \sqrt{11})$

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Similar questions

Show that the are of the triangle formed by the lines $y = m_{1} x, y = m_{2}$ x and y = c is equal to $\frac{c^{2}}{4} (\sqrt{33} + \sqrt{11})$ , where $m_{1}, m_{2}$ are the roots of the equation $x^{2} + (\sqrt{3} + 2) x + \sqrt{3} - 1 = 0.$

Q. If

m_{1}

and

m_{2}

are the roots of the equation

x^{2} + (\sqrt{3} + 2) x + (\sqrt{3} - 1) = 0

, then the area of the triangle formed by the lines

y = m_{1} x, y = m_{2} x

and

y = 2

is :

Q. Show that the area of the triangle formed by the lines

y = m_{1} x, y = m_{2} x a n d y = c

is equal to

\frac{c^{2}}{4} (\sqrt{33} + \sqrt{11}),

where

m_{1}, m_{2}

are the roots of the equation

x^{2} + (\sqrt{3} + 2) x + \sqrt{3} - 1 = 0

Q. If

m_{1}

and

m_{2}

are the roots of the equation

x^{2} + (\sqrt{3} + 2) x + (\sqrt{3} - 1) = 0,

then the area of the triangle formed by the lines

y = m_{1} x,

y = m_{2} x

and

y = 2,

Q. A triangle has the lines

y = m_{1} x

and

y = m_{2} x

for two of its sides, where

m_{1}, m_{2}

are the roots of the equation

x^{2} + a x - 1 = 0

, then find the orthocentre of triangle.

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Show that the area of the triangle formed by the lines y = m1 x, y = m2 x and y = c is equal to c2433+11, where m1, m2 are the roots of the equation x2+3+2x+3-1=0.

Show that the area of the triangle formed by the lines y = m₁ x, y = m₂ x and y = c is equal to $\frac{c^{2}}{4} (\sqrt{33} + \sqrt{11}),$ where m₁, m₂ are the roots of the equation $x^{2} + (\sqrt{3} + 2) x + \sqrt{3} - 1 = 0 .$