Family of Straight Lines

Q.

If the lines represented by the equation $a{x}^2-bxy-y^2=0$ make angles $\alpha$ and $\beta$ with the $x-$axis, then $\tan \left(\alpha +\beta \right)$.

1. $\frac{b}{1+a}$

2. $-\frac{b}{1+a}$

3. $\frac{a}{1+b}$

4. None of these

Q.

A general equation of second degree $a{x}^2+2hxy+b{y}^2+2gx+2fy+c=0$ represents a pair of straight lines if.

1. $h^2=2ab$

2. $h^2>2ab$

3. $\left|\begin{array}{ccc}a& h& g\\ h& b& f\\ g& f& c\end{array}\right|=0$

4. None of these

Q.

If the sum of slopes of the pair of lines represented by $4x^2+2hxy-7y^2=0$ is equal to the product of the slopes, then the value of $h$ is:

1. $–6$

2. $–2$

3. $–4$

4. $4$

Q.

If the equation $hxy+gx+fy+c=0$represents a pair of straight lines, then

1. $fh=cg$

2. $fg=ch$

3. $h^2=gf$

4. $fgh=c$

Q. Let line $L$ passes through the point of intersection of $2x+y-1=0$ and $x+2y-2=0$. If $L$ makes a triangle with the coordinate axes of area $\frac{5}{8}\text{sq. units}$, then the equation of $L$ can be

1. $4x-5y+5=0$
2. $4x+5y-5=0$
3. $4x-5y-5=0$
4. $4x+5y+5=0$

Q. $\text{List I}$ has four entries and $\text{List II}$ has five entries. Each entry of $\text{List I}$ is to be matched with one entry of $\text{List II}.$

$\begin{array}{ccccc}& \text{List I}& & \text{List II}& \\ \text{(A)}& \text{If}x=\sqrt{6+\sqrt{6+\sqrt{6+\dots \text{up to}\infty }}},& & & \\ & \text{then}x\text{is equal to}& \text{(P)}& 4& \\ \text{(B)}& \text{If}a\text{and}x\text{are positive integers such}& & & \\ & \text{that}x<a\text{and}\sqrt{a-x},\sqrt{x},\sqrt{a+x}& \text{(Q)}& 5& \\ & \text{are in}A.P.\text{, then least possible value of}a\text{is}& & & \\ \text{(C)}& \text{If}3a+2b+4c=0,a,b,c\in R\text{and the}& & & \\ & \text{line}ax+by+c=0\text{always passes}& & & \\ & \text{through a fixed point}(p,q)\text{, then the}& & & \\ & \text{value of}2p+q\text{is}& \text{(R)}& 2& \\ \text{(D)}& \text{If}k(\sin 18{}^{\circ }+\cos 36{}^{\circ }{)}^2=5,\text{then the}& & & \\ & \text{value of}k\text{is}& \text{(S)}& 3& \\ & & \text{(T)}& 6& \end{array}$

Which of the following is the only CORRECT combination?

1. $A\to R;B\to S$
2. $A\to S;B\to Q$
3. $A\to Q;B\to S$
4. $A\to S;B\to R$

Q. The locus of the orthocentre of the triangle formed by the lines $(1+p)x-py+p(1+p)=0,$$(1+q)x-qy+q(1+q)=0$ and $y=0,$ where $p\ne q,$ is

1. $y=-2x$
2. $y=2x$
3. $y=x$
4. $y=-x$

Q. In a rhombus $ABCD,$ the slope of diagonal $AC$ is $1$ and is among the family of lines $(x+2y-5)+\lambda (3x+y-5)=0,$ where $\lambda \in R.$ One of the vertex of the rhombus is $(-2,3)$ and the area of rhombus is $10\sqrt{2}$ sq. units, then which of the following is/are correct?

1. Length of smaller diagonal is $5$
2. Length of larger diagonal is $6\sqrt{2}$
3. One of the vertex is $(2,-1)$
4. Perimeter of rhombus is $2\sqrt{57}$ units

Q. If $a,b,c$ are in A.P., then the line $ax+by+c=0$ will always pass through a fixed point $(h,k)$, then the value of $h-k$ is

Q. The equation of the straight line passing through the point of intersection of the lines $x-y=1$ and $2x-3y+1=0$ and parallel to the line $3x+4y=14$ is

1. $3x+4y+24=0$
2. $3x+4y-24=0$
3. $4x+3y+24=0$
4. $4x+3y-24=0$

Q. The family of lines $x(a+b)+y(a-b)=2a,a,b\in R$ are concurrent at $(p,q)$ then the value of $p+q$ is

Q. Let $y=mx+{\lambda }_i,i=1,2,3,...,n$ be a family of $n$ parallel lines subjected to following conditions.
$\left(1\right)m$ being a constant
$\left(2\right)\sum _{i=1}^n{\lambda }_i=1$
A variable line through origin intersects the lines at $P_i(i=1,2,3,...,n)$ and $Q$ be a point on variable line such that $\sum _{i=1}^{n}O{P}_i=OQ$. If the locus of $Q$ is a straight line which passes through a fixed point $(a,b)\forall m\in R,$ then the value of $(3a+2b)$ is

1. $1$
2. $2$
3. $5$
4. $10$

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

1. $33x+12y-7=0$
2. $33x+12y+7=0$
3. $12x+33y-7=0$
4. $12x+33y+7=0$

Q. Given the family of lines $a(3x+4y+6)+b(x+y+2)=0.$ The line of the family situated at the greatest distance from the point $P(2,3),$ has the equation

1. $4x+3y+8=0$
2. $5x+3y+10=0$
3. $15x+8y+30=0$
4. $4x-3y+8=0$

Q. If the lines $x+y=|a|$ and $ax-y=1$ intersect each other in the first quadrant, then the range of $a$ is

1. $(1,\infty )$
2. $(-1,\infty )$
3. $(-1,1)$
4. $(0,\infty )$

Q. The equation of the straight line through the intersection of line $2x+y=1$ and $3x+2y=5$ and passes through the origin is

1. $7x+3y=0$
2. $7x-y=0$
3. $3x+2y=0$
4. $3x+y=0$

Q. The equation of the straight line through the intersection of line $2x+y=1$ and $3x+2y=5$ and passes through the origin is

1. $7x+3y=0$
2. $7x-y=0$
3. $3x+2y=0$
4. $3x+y=0$

Q. If the coordinates of the four vertices of a quadrilateral are $(-2,4),(-1,2),(1,2)$ and $(2,4)$ taken in order, then the equation of line passing through the vertex $(-1,2)$ and dividing the quadrilateral in two equal areas is

1. $x+1=0$
2. $x+y-1=0$
3. $x-y+3=0$
4. $x-y-3=0$

Q. $ABCD$ is a rhombus. The slope of $AC$ is $1$ and is among the family of lines $(x+2y-5)+\lambda (3x+y-5)=0,$ where $\lambda \in R.$ One of the vertex of the rhombus is $(-2,3).$ If the area of rhombus is $10\sqrt{2}$ sq. units, then which of the following is (are) CORRECT?

1. Length of smaller diagonal is $4$
2. Length of larger diagonal is $4\sqrt{2}$
3. One of the vertex is $(2,-1)$
4. Perimeter of rhombus is $2\sqrt{57}$

Q. If the straight line $x(a+2b)+y(a+3b)=a+b$ passes through a fixed point for different values of $a$ and $b$, then the fixed point is

1. $(2,-1)$
2. $(-2,1)$
3. $(-1,2)$
4. $(1,-2)$

Q. $ABCD$ is a rhombus. The slope of $AC$ is $1$ and is among the family of lines $(x+2y-5)+\lambda (3x+y-5)=0,$ where $\lambda \in R.$ One of the vertex of the rhombus is $(-2,3).$ If the area of rhombus is $10\sqrt{2}$ sq. units, then which of the following is (are) CORRECT?

1. Length of smaller diagonal is $4$
2. Length of larger diagonal is $4\sqrt{2}$
3. One of the vertex is $(2,-1)$
4. Perimeter of rhombus is $2\sqrt{57}$

Q.

Equation of the straight line passing through the point of intersection of the lines $3x+4y=7,x-y+2=0$ and having slope $3$ is

1. 21x−7y+16=0

2. 9x−3y+14=0

3. 21x−7y+12=0

4. 9x−3y+5=0

Q.

Find the point of intersection for the following two lines: $\left\{\begin{array}{l}x+y=5\\ 3x+y=1\end{array}\right.$

Q. Locus of the image of the point (2, 3) in the line $(2x-3y+4)+k(x-2y+3)=0,kϵR,$ is a

1. Straight line parallel to x-axis
2. Straight line parallel to y-axis
3. Circle of radius $\sqrt{2}$
4. Circle of adius 3

Q. The number of integeral values of $m,$ for which the $x$ -coordinate of the point of intersection of the lines $3x+4y=9$ and $y=mx+1$ is an integer is

1. $2$
2. $0$
3. $4$
4. $1$

Q. Consider a family of straight lines $(x+y)+\lambda (2x-y+1)=0$. Then the equation of the straight line belonging to this family that is farthest from $(1,-3)$ is:

1. $6x+15y+5=0$
2. $6x-15y+5=0$
3. $6x+15y+7=0$
4. $6x-15y+7=0$

Q. If the lines $x+y=|a|$ and $ax-y=1$ intersect each other in the first quadrant, then the range of $a$ is

1. $(1,\infty )$
2. $(-1,\infty )$
3. $(-1,1)$
4. $(0,\infty )$