(C) The given equation is $(x - 2y)^2 + k(x - 2y) = 0$.This can be factored as $(x - 2y)(x - 2y + k) = 0$.Thus,the two lines are $L_1: x - 2y = 0$ and

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(C) The given equation is $(x - 2y)^2 + k(x - 2y) = 0$.This can be factored as $(x - 2y)(x - 2y + k) = 0$.Thus,the two lines are $L_1: x - 2y = 0$ and $L_2: x - 2y + k = 0$.The distance $d$ between two parallel lines $ax + by + c_1 = 0$ and $ax + by + c_2 = 0$ is given by $d = \frac{|c_1 - c_2|}{\sqrt{a^2 + b^2}}$.Here,$a = 1$,$b = -2$,$c_1 = 0$,and $c_2 = k$.So,$3 = \frac{|0 - k|}{\sqrt{1^2 + (-2)^2}} = \frac{|k|}{\sqrt{1 + 4}} = \frac{|k|}{\sqrt{5}}$.Therefore,$|k| = 3\sqrt{5}$,which implies $k = \pm 3\sqrt{5}$.

Class 11 Mathematics Pair of straight lines Equation of lines joining the origin to the point of intersection of a curve and a line and Distance between the pair of lines

Medium

Two lines are given by $(x - 2y)^2 + k(x - 2y) = 0$. The value of $k$ so that the distance between them is $3$,is

A
$1/\sqrt{5}$
B
$\pm 2/\sqrt{5}$
C
$\pm 3\sqrt{5}$
D
None of these

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Pair of straight lines

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Equation of lines joining the origin to the point of intersection of a curve and a line and Distance between the pair of lines

The pair of straight lines joining the origin to the points of intersection of the line $y = 2\sqrt{2}x + c$ and the circle $x^2 + y^2 = 2$ are at right angles,if

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$A$ pair of lines drawn through the origin forms a right-angled isosceles triangle with the line $2x + 3y = 6$,having the right angle at the origin. The area (in sq. units) of the triangle thus formed is

MediumAP EAMCET 2024

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The area (in sq. units) of the triangle formed by the lines $x^2-3xy+y^2=0$ and $x+y+1=0$ is

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If $\theta$ is the acute angle between the lines joining the origin to the points of intersection of the curve $x^2+xy+y^2+x+3y+1=0$ and the straight line $x+y+2=0$,then $\cos \theta=$

EasyTS EAMCET 2023

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$A$ pair of perpendicular lines passes through the origin and also through the points of intersection of the curve $x^2+y^2=4$ with $x+y=a$,where $a>0$. Then $a$ is equal to

DifficultTS EAMCET 2010

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Two lines are given by $(x - 2y)^2 + k(x - 2y) = 0$. The value of $k$ so that the distance between them is $3$,is

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The pair of straight lines joining the origin to the points of intersection of the line $y = 2\sqrt{2}x + c$ and the circle $x^2 + y^2 = 2$ are at right angles,if

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