Condition for Concurrency of Three Lines

Q.

A quadrilateral has the vertices at the points $(-4,2),(2,6),(8,5)and(9,-7)$. Show that the mid points of the sides of this quadrilateral are the vertices of a parallelogram by using slope formula.

Q.

Three lines $px+qy+r=0,qx+ry+p=0$ and $rx+py+q=0$ are concurrent, if

1. $p+q+r=0$
2. $p^2+q^2+r^2=pr+rq$
3. $p^3+q^3+r^3=3pqr$
4. None of these

Q. The length of altitude through $A$ of the triangle $ABC$, where $A=(-3,0);B=(4,-1);C=(5,2).$

1. $\frac{2}{\sqrt{10}}$
2. $\frac{11}{\sqrt{10}}$
3. $\frac{4}{\sqrt{10}}$
4. $\frac{22}{\sqrt{10}}$

Q. $PS$ is a median and $QL$ and $RM$ are perpendiculars drawn from $Q$ and $R$ respectively on $PS$ produced. Then which of the following statements is correct$?$

1. $PQ=RM$
2. $QL=RM$
3. $PL=SR$
4. $PS=SM$

Q.

Consider the lines given by
$L_1:x+3y-5=0$
$L_2:3x-ky-1=0$
$L_3:5x+2y-12=0$

Match the Statements/Exp[ressions in Column I with the Statements/Expressions in Column II.
$\begin{array}{cc}\text{Column I}& \text{Column II}\\ L_1,L_2,L_3\text{are concurrent, if}& \text{k = -9}\\ \text{One of}{L}_1,L_2,L_3\text{is parallel to at least one of the other two, if}& \text{k}=\frac{6}{5}\\ L_1,L_2,L_3\text{form a triangle, if}& k=\frac{5}{6}\\ L_1,L_2,L_3\text{do not form a triangle, if}& \text{k = 5}\end{array}$

1. (a) $\to$ (s), (b) $\to$ (p, q), (c) $\to$ (r), (d) $\to$ (p, q, s)

2. (a) $\to$ (p), (b) $\to$ (p, q, s), (c) $\to$ (r), (d) $\to$ (p, q)

3. (a) $\to$ (s), (b) $\to$ (p, q, s), (c) $\to$ (s, r), (d) $\to$ (p, q, s)

4. none of these

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 straight lines 2x+3y-12 = 0, x+y-5 = 0 and x-2y+1 = 0 are concurrent.

1. True

2. False

Q.

If 4a+5b+6c=0 then the set of lines ax+by+c=0 are concurrent at the point

1. 

2. 

3. 

4. 

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.

Consider the lines given by
$L_1:x+3y-5=0$
$L_2:3x-ky-1=0$
$L_3:5x+2y-12=0$

Match the Statements/Exp[ressions in Column I with the Statements/Expressions in Column II.
$\begin{array}{cc}\text{Column I}& \text{Column II}\\ L_1,L_2,L_3\text{are concurrent, if}& \text{k = -9}\\ \text{One of}{L}_1,L_2,L_3\text{is parallel to at least one of the other two, if}& \text{k}=\frac{6}{5}\\ L_1,L_2,L_3\text{form a triangle, if}& k=\frac{5}{6}\\ L_1,L_2,L_3\text{do not form a triangle, if}& \text{k = 5}\end{array}$

1. (a) $\to$ (s), (b) $\to$ (p, q), (c) $\to$ (r), (d) $\to$ (p, q, s)

2. (a) $\to$ (p), (b) $\to$ (p, q, s), (c) $\to$ (r), (d) $\to$ (p, q)

3. (a) $\to$ (s), (b) $\to$ (p, q, s), (c) $\to$ (s, r), (d) $\to$ (p, q, s)

4. none of these