Condition for Concurrency of Three Lines

Q. If two distinct chords, drawn from the point(p, q) on the circle $x^2+y^2=px+qy(wherepq\ne 0)$ are bisected by the x-axis, then

1. 
2. 
3. <
4. >

Q. 28. Solve for real x for the inequality by wavy curve method: (x-5)(x-9)÷{(x-3)(x-7)} <_ 0 (less than or equal to zero)

Q.

If the line joining origin to the points of intersection of the line fx - gy = $\lambda$ and the curve $x^2+hxy-y^2+gx+fy=0$ be mutually perpendicular, then

1. 

2. 

3. 

4. 

Q. 17. Let 1/(a1-2i) , 1/(a2-2i) , 1/(a3-2i) , 1/(a4-2i), 1/(a5-2i), 1/(a6-2i), 1/(a7-2i), 1/(a8-2i) are vertices of regular octagon. If the area of octagan is A. Then the value of 82A is ( where aj belongs to R for j=1, 2, 3, 4, 5, 6, 7, 8 and i=-1 )

Q.

POQis a straight line through the origin O, P and Q represent the complex numbers a+ib andc+id respectively and OP=OQ, then

1. 

2. arg(a+ib)=arg(c+id)

3. None of these

4. a+c=b+d

Q. If three lines whose equations are concurrent, then show that

Q.

If the line x=a+m, y=-2 and y=mx are concurrent , then least value of |a| is

1. 0

2. 

3. 

4. 2

Q. Consider the lines represented by the equation $(x^2+xy-x)(x-y)=0$, forming a triangle, then
$\begin{array}{cc}Column1& Column2\\ \left(a\right)Orthocentreoftraingle& (\frac{1}{6},\frac{1}{2})\\ \left(b\right)Circumcenter& \left(\frac{1}{2+2\sqrt{2}}\right)\\ \left(c\right)Centroid& (0,\frac{1}{2})\\ \left(d\right)Incenter& (\frac{1}{2},\frac{1}{2})\end{array}$

1. a-s, b-r, c-p, d-q

2. b-r, b-s, c-q, d-p

3. a-r, b-s, c-p, d-q

4. a-s, b-r, c-q, d-p1

Q.

If three lines whose equations are $y=m_{1}x+c_1,y=m_{2}x+c_2$ and $y=m_{3}x+c_3$ are concurrent, then show that $m_1(c-2-c_3)+m_2(c_3-c_1+m_3(c_1-c_2)=0$.

Q.

A rod of length 12 cm moves with its ends always touching the coordinate axes. Determine the equation of the path of a moving point P on the rod which is 3 cm from the end in contact with the x-axis.

Q. Find equation of the line passing through $(0,0)$ with slope $m$.

Q. Find the coordinates of the point which divides the line segment joining the points $(-2,3,5)$ and $(1,-4,6)$ in the ratio
(i) $2:3$ internally
(ii) $2:3$ externally

Q. Find the value of p so that the three lines 3 x + y – 2 = 0, px + 2 y – 3 = 0 and 2 x – y – 3 = 0 may intersect at one point.

Q. 47. The combined equation of lines passing through (1, 1) making an angle π/3 with the line x-2y+1=0 is

Q. Find the value of $x$ for which the points $(x,-1),(2,1)$ and $(4,5)$ are collinear.

Q.

Three lines

$L_1$ = x-y+6 = 0

$L_2$ = 2x+y-3 = 0

$L_3$ = x-2y+m = 0 are given. Which of the following can be the value of m if $L_1$_, $L_2$ and $L_3$ form a triangle.

1. 10

2. 11

3. 12

4. 13

Q. Let $ABC$ and $PQR$ be any two triangles in the same plane. Assume that the perpendiculars from the points $A,B,C$ to the sides $QR,RP,PQ$ respectively are concurrent. Using vector methods or otherwise, prove that the perpendiculars from $P,Q,R$ to $BC,CA,AB$ respectively are also concurrent.

Q. The mid-point of the points on $x+y=4$ which are at a unit distance from the line $4x+3y-10=0$ is

1. (-2, 6)
2. (2, 2)
3. (-7, 11)
4. (1, 3)

Q. Find the distance of the line $4x+7y+5$ $=$ $0$ from the point $(1,2)$ along the line $2x-y=$ $0$.

Q. Find the values of k for which the line$(k-3)x-(4-k^2)y+k^2-7k+6=0$ is :
(a) Parallel to the $x$-axis.
(b) Parallel to the $y$-axis.
(c) Passing through the origin.

Q.

The straight lines x+2y-9=0, 3x+5y-5=0 and ax+by-1=0 are concurrent if the straight line 22x-35y-1=0 passes through the point

1. (a, b)

2. (b, a)

3. (−a, b)

4. (−a, −b)

Q. If the equation of line parallel to the line $3x-4y+2=0$ and passing through the point $(-2,3)$ is $3x-4y+k,$ value of $k$ is

Q. Find the equation of the line perpendicular distance from the origin is $5$ units and the angle made by the perpendicular with the positive $x$-axis is $30{}^{\circ }$.

Q.

The condition for three lines $a_1$x+$b_1$y+$c_1$ = 0, $a_2$x+$b_2$y+$c_2$ = 0 and $a_3$x+$b_3$y+$c_3$ = 0 , to be concurrent is

$\left|\begin{array}{ccc}{a}_1& b_1& c_1\\ a_2& b_2& c_2\\ a_3& b_3& c_3\end{array}\right|=0$

1. True

2. False

Q. Find the equation of a line that cuts off equal intercepts on the coordinate axes and passes through the point $(2,3)$.

Q. The perpendicular from the origin to the line $y$ $=mx+c$ meets it at the point $(-1,2)$. Find the values of $m$ and $c$.

Q. Find the equation of a line drawn perpendicular to the line$\frac{x}{4}+\frac{y}{6}=1$ through the point, where it meets the $y$-axis.

Q. Find the equation of the line intersecting the $x$-axis at a distance of $3$ units to the left of origin with slope $-2.$

Q. The variance of first $n$ natural numbers, is

1. $\frac{n+1}{2}$
2. $\frac{n^2-1}{6}$
3. $\frac{n^2+1}{12}$
4. $\frac{n^2-1}{12}$

Q. Find the equation of the line passing through the point of intersection of the lines$4x-7y-3=0$ and $2x-3y+1=0$ that has equal intercepts on the axes.