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GradeFamily Of Straight Lines

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If $ \dfrac{a}{{\sqrt {bc} }} - 2 = \sqrt {\dfrac{b}{c}} + \sqrt {\dfrac{c}{b}} $ , where $ a,b,c > 0 $ then family of lines $ \sqrt a x + \sqrt b y + \sqrt c = 0 $ passes through the fixed point given by

A. $ \left( {1,1} \right) $

B. $ \left( {1, - 2} \right) $

C. $ \left( { - 1,2} \right) $

D. $ \left( { - 1,1} \right) $

A. $ \left( {1,1} \right) $

B. $ \left( {1, - 2} \right) $

C. $ \left( { - 1,2} \right) $

D. $ \left( { - 1,1} \right) $

If \[{a^2} + {b^2} - {c^2} - 2ab = 0\], then the given family of straight lines \[ax + by + c = 0\], is concurrent at the points?

A. (-1,1)

B. (1,-1)

C. (1,1)

D. (-1,-1)

A. (-1,1)

B. (1,-1)

C. (1,1)

D. (-1,-1)

In a plane there are two families of lines $y=x+r,$ $y=-x+r,$ where $r\in 0,1,2,3,4.$ The number of squares of diagonals of the length 2 units formed by the lines is

Consider the family of lines \[(x + y - 1) + \lambda (2x + 3y - 5) = 0\] and \[(3x + 2y - 4) + \mu (x + 2y - 6) = 0\]. Find the equation of the straight line that belongs to both the families.

A. \[x - 2y - 8 = 0\]

B. \[x - 2y + 8 = 0\]

C. \[2x + y - 8 = 0\]

D. \[2x + y - 8 = 0\]

A. \[x - 2y - 8 = 0\]

B. \[x - 2y + 8 = 0\]

C. \[2x + y - 8 = 0\]

D. \[2x + y - 8 = 0\]

If the family of lines $x\left( a+2b \right)+y\left( 3a+b \right)=a+b$ passes through the point for all values of $a$ and $b$, then the coordinates of the points is:\[\]

A.$\left( 2,1 \right)$\[\]

B.$\left( 2,-1 \right)$\[\]

C.$\left( -2,1 \right)$\[\]

D. None of these \[\]

A.$\left( 2,1 \right)$\[\]

B.$\left( 2,-1 \right)$\[\]

C.$\left( -2,1 \right)$\[\]

D. None of these \[\]

Family of lines represented by the equation $\left( \cos \theta +\sin \theta \right)x+\left( \cos \theta -\sin \theta \right)y-3\left( 3\cos \theta +\sin \theta \right)=0$ passes through fixed point $M$ for all real value of $\theta $. Find $M$

A. $\left( 6,3 \right)$

B. $\left( 3,6 \right)$

C. $\left( -6,2 \right)$

D. $\left( 3,-6 \right)$

A. $\left( 6,3 \right)$

B. $\left( 3,6 \right)$

C. $\left( -6,2 \right)$

D. $\left( 3,-6 \right)$

The family of straight lines \[3\left( a+1 \right)x-4\left( a-1 \right)y+3\left( a+1 \right)=0\] for different values of\['a'\] passes through a fixed point whose co – ordinates are

(a) \[\left( 1,0 \right)\]

(b) \[\left( -1,0 \right)\]

(c) \[\left( -1,-1 \right)\]

(d) None of the above.

(a) \[\left( 1,0 \right)\]

(b) \[\left( -1,0 \right)\]

(c) \[\left( -1,-1 \right)\]

(d) None of the above.

Locus of the perpendicular lines one belonging to family \[\left( x+y-2 \right)+\lambda \left( 2x+3y-5 \right)=0\] and other belonging to family \[\left( 2x+y-11 \right)+\lambda \left( x+2y-13 \right)=0\] is a

(a) Circle

(b) Straight line

(c) Pair of lines

(d) None

(a) Circle

(b) Straight line

(c) Pair of lines

(d) None

Family of the lines $x{\sec ^2}\theta + y{\tan ^2}\theta - 2 = 0$, for different real $\theta $, is:

(A) Not concurrent

(B) Concurrent at $\left( {1,1} \right)$

(C) Concurrent at $\left( {2, - 2} \right)$

(D) Concurrent at $\left( { - 2,2} \right)$

(A) Not concurrent

(B) Concurrent at $\left( {1,1} \right)$

(C) Concurrent at $\left( {2, - 2} \right)$

(D) Concurrent at $\left( { - 2,2} \right)$

Given the family of lines, $(3x + 4y + 6) + \lambda (x + y + 2) = 0$. The line of the family is situated at the greatest distance from the point $P\left( {2,3} \right)$ has equation:

A) $4x + 3y + 8$

B) $5x + 3y + 10 = 0$

C) $15x + 8y + 30 = 0$

D) None of these

A) $4x + 3y + 8$

B) $5x + 3y + 10 = 0$

C) $15x + 8y + 30 = 0$

D) None of these

A family of lines is given by $\left( 1+2\lambda \right)x+\left( 1-\lambda \right)y+\lambda =0$ , $\lambda $ being the parameter. The line belonging to this family at the maximum distance from the point $\left( 1,4 \right)$ is

(a)$4x-y+1=0$

(b)$33x+12y+7=0$

(c)$12x+33y=7$

(d)None of these

(a)$4x-y+1=0$

(b)$33x+12y+7=0$

(c)$12x+33y=7$

(d)None of these

The two diagonally opposite vertices of a square are (6,6) and (0,0). Find the point which lies on X-axis.

$

{\text{A}}{\text{. }}\left( {6,0} \right) \\

{\text{B}}{\text{. }}\left( {0,6} \right) \\

{\text{C}}{\text{. }}\left( {6,6} \right) \\

{\text{D}}{\text{. None}} \\

$

$

{\text{A}}{\text{. }}\left( {6,0} \right) \\

{\text{B}}{\text{. }}\left( {0,6} \right) \\

{\text{C}}{\text{. }}\left( {6,6} \right) \\

{\text{D}}{\text{. None}} \\

$

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