One of the lines denoted by $L_{1}$ coincides with one of the lines in $L_{2}$ . Let the other lines from $L_{1}, L_{2}$ together be denoted by line pair $L_{3}$ . The centroid of the triangle formed by the pair of lines given by $L_{3}$ and $x = 1$ is

(\frac{2}{3}, - \frac{2}{3})

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(\frac{2}{3}, - 1)

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(\frac{1}{3}, - \frac{1}{3})

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(\frac{2}{3}, - \frac{1}{3})

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Solution

The correct option is B $(\frac{2}{3}, - \frac{1}{3})$
$L_{1} : 6 x^{2} + x y - y^{2} = 0 \Rightarrow (3 x - y) (2 x + y) = 0$
If $2 x + y$ coincides with $L_{2} : 3 x^{2} - a x y + y^{2} = 0$ then $3 + 2 a + 4 = 0 \Rightarrow a = - 7 / 2$ ,
which is not possible as $a > 0$
So if $y = 3 x$ coincides with $L_{2}$ then $3 - 3 a + 9 = 0 \Rightarrow a = 4$
$L_{2} : 3 x^{2} - 4 x y + y^{2} = 0 \Rightarrow (3 x - y) (x - y) = 0 a n d L_{3} : (2 x + y) (x - y) = 0$
$p_{1} = \pm \frac{16 + 4}{\sqrt{5}} = \pm 4 \sqrt{5}, p_{2} = \pm \frac{8 - 4}{\sqrt{2}} = \pm 2 \sqrt{2}$
$\Rightarrow ∣ ∣ p_{1}^{2} - p_{2}^{2} ∣ ∣ = 80 - 8 = 72$
Vertices of the triangle formed by the lines $L_{3}$ and $x = 1$ are $(0, 0), (1, - 2) . (1, 1) .$
So the coordinates of the centroid of the triangle are $(\frac{2}{3}, - \frac{1}{3})$ .

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Find the equation of the circle circumscribing the triangle formed by the lines $L_{1} = 0, L_{2} = 0$ and $L_{3} = 0$

Q. Line

l_{1}

is perpendicular to line

l_{2}

and line

l_{2}

is perpendicular to line

l_{3 .}

Then, what can be said about lines

l_{1}

and

l_{3 .}

Consider the lines given by
$L_{1} : x + 3 y - 5 = 0$
$L_{2} : 3 x - k y - 1 = 0$
$L_{3} : 5 x + 2 y - 12 = 0$

Match the Statements/Exp[ressions in Column I with the Statements/Expressions in Column II.
$\begin{matrix} Column I & Column II L_{1}, L_{2}, L_{3} are concurrent, if & k = -9 One of L_{1}, L_{2}, L_{3} is parallel to at least one of the other two, if & k = \frac{6}{5} L_{1}, L_{2}, L_{3} form a triangle, if & k = \frac{5}{6} L_{1}, L_{2}, L_{3} do not form a triangle, if & k = 5 \end{matrix}$

Q. Coordinates of the points of intersection of the lines given by

L_{1}, L_{2}

and

L_{3}

with

x^{2} + y^{2} = 5

are

Q. Consider the lines given by

L_{1} : x + 3 y - 5 = 0

L_{2} : 3 x - k y - 1 = 0

L_{3} : 5 x + 2 y - 12 = 0

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One of the lines denoted by L1 coincides with one of the lines in L2. Let the other lines from L1,L2 together be denoted by line pair L3. The centroid of the triangle formed by the pair of lines given by L3 and x=1 is

One of the lines denoted by $L_{1}$ coincides with one of the lines in $L_{2}$ . Let the other lines from $L_{1}, L_{2}$ together be denoted by line pair $L_{3}$ . The centroid of the triangle formed by the pair of lines given by $L_{3}$ and $x = 1$ is