General Form of a Straight Line

Q. Consider a triangle $\Delta$ whose two sides lie on the $x-$axis and the line $x+y+1=0.$ If the orthocenter of $\Delta$ is $(1,1),$ then the equation of the circle passing through the vertices of the triangle $\Delta$ is

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

Q. Consider the system of linear equations
$-x+y+2z=0$
$3x-ay+5z=1$
$2x-2y-az=7$
Let $S_1$ be the set of all $a\in R$ for which system is inconsistent and $S_2$ be the set of all $a\in R$ for which the system has infinitely many solutions. If $n\left(S_1\right)$ and $n\left(S_2\right)$ denote the number of elements in $S_1$ and $S_2$ respectively, then

1. $n\left(S_1\right)=2,n\left(S_2\right)=0$
2. $n\left(S_1\right)=1,n\left(S_2\right)=0$
3. $n\left(S_1\right)=2,n\left(S_2\right)=2$
4. $n\left(S_1\right)=0,n\left(S_2\right)=2$

Q.

The angle between the lines represented by the equation $\left(x^2+y^2\right)\sin \left(\theta \right)+2xy=0$ is

1. $\theta$

2. $\frac{\theta }{2}$

3. $\frac{\mathrm{\pi }}{2}-\theta$

4. $\frac{\mathrm{\pi }}{2}-\frac{\theta }{2}$

Q. The equation of the straight line passing through the point $(4,3)$ and making intercepts on the co-ordinate axes whose sum is $-1$ is

1. $\frac{x}{2}-\frac{y}{3}=1$
2. $\frac{x}{-2}+\frac{y}{1}=1$
3. $-\frac{x}{3}+\frac{y}{2}=1$
4. $\frac{x}{1}-\frac{y}{2}=1$

Q. Let $A$ be the set of all points $(\alpha ,\beta )$ such that the area of triangle formed by the points $(5,6),(3,2)$ and $(\alpha ,\beta )$ is $12$ square units. Then the least possible length of a line segment joining the origin to a point in $A,$ is

1. $\frac{8}{\sqrt{5}}$
2. $\frac{16}{\sqrt{5}}$
3. $\frac{4}{\sqrt{5}}$
4. $\frac{12}{\sqrt{5}}$

Q.

The equation of the line passing through the point of intersection of the lines $x-3y+2=0$ and $2x+5y-7=0$ and perpendicular to the line $3x+2y+5=0$, is

1. $2x–3y+1=0$

2. $6x–9y+11=0$

3. $2x–3y+15=0$

4. $3x–2y+1=0$

Q.

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

1. $2$

2. $0$

3. $4$

4. $1$

Q. Equation of the hour hand at 4 O’ clock is

1. $x-\sqrt{3}y=0$
2. $\sqrt{3}x-y=0$
3. $x+\sqrt{3}y=0$
4. $\sqrt{3}x+y=0$

Q.

Find the equations of the two lines through the origin which intersect the line $\frac{x-3}{2}=\frac{y-3}{1}=\frac{z}{1}$ at angles of $\frac{\pi }{3}$ each.

Q. In a regular hexagon $ABCDEF,\overrightarrow{AE}=$

1. $\overrightarrow{AC}+\overrightarrow{AF}+\overrightarrow{AB}$
2. $\overrightarrow{AC}+\overrightarrow{AF}-\overrightarrow{AB}$
3. $\overrightarrow{AC}+\overrightarrow{AB}-\overrightarrow{AF}$
4. $\overrightarrow{AC}+\overrightarrow{AB}-\overrightarrow{AD}$

Q. Let the two vertices of a triangle are $(2,-1)$ and $(3,2)$ and third vertex lies on the line $x+y=5$. If the area of triangle is $4\text{sq. units}$, then the coordinates of the third vertex is/are

1. $(0,5)$
2. $(5,0)$
3. $(4,1)$
4. $(1,4)$

Q. A ray of light is incident along a line which meets another line, $7x–y+1=0$, at the point $(0,1)$. The ray is then reflected from this point along the line, $y+2x=1$. Then the equation of the line of incidence of the ray of light is :

1. $41x+38y–38=0$
2. $41x+25y–25=0$
3. $41x–25y+25=0$
4. $41x-38y+38=0$

Q. The equation of the line parallel to $Y-$axis and $3$units to the right of it, is

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

Q.

Let

$A+2B=\left[\begin{array}{ccc}1& 2& 0\\ 6& -3& 3\\ -5& 3& 1\end{array}\right]$

and

$2A-B=\left[\begin{array}{ccc}2& -1& 5\\ 2& -1& 6\\ 0& 1& 2\end{array}\right]$

If $Tr\left(A\right)$ denotes the sum of all diagonal elements of the matrix $A$, then $Tr\left(A\right)-Tr\left(B\right)$ has value equal to:

1. $0$

2. $1$

3. $3$

4. $2$

Q. A line through A (-5, -4) meets the lines x + 3y + 2 = 0, 2x + y + 4 = 0, 2x + y + 4 = 0 and x – y – 5 = 0 at B, C and D respectively. If ${\left(\frac{15}{AB}\right)}^2+{\left(\frac{10}{AC}\right)}^2={\left(\frac{6}{AD}\right)}^2,$, then the equation of the line is

1. 3x -2y + 3 = 0
2. None of these
3. 2x + 3y + 22 = 0
4. 5x – 4y + 7 = 0

Q.

The position of the points $\left(3,4\right)$ and $(2,-6)$ with respect to the line $3x-4y=8$ are

1. On the same side of the line

2. On different side of the line

3. One point on the line and the other outside the line

4. Both points on the line

Q. The line $\frac{x}{a}+\frac{y}{b}=1$ cuts the axis at $A$ and $B$, another line perpendicular to $AB$ cuts the axes at$P,Q$respectively.Locus of points of intersection of $AQ$ and $BP$ is

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

Q. The equation of the line parallel to $Y-$axis and $3$units to the right of it, is

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

Q.

Find the equation of the line passing through the point of intersection of $2x-7y+11=0$ and $x+3y-8=0$ and is parallel to (i) x-axis (ii) y-axis.

Q. 12.Order and degree of the differential equation of family of lines situated at a constant distance p from the origin?

Q.

Find the equation of the straight lines passing through the origin and making an angle of ${45}^{\circ }$ with the straight line $\sqrt{3}x+y=11$

Q.

If ${}^{10}C_x={}^{10}C_{x+4}$, then prove that $x=3$.

Q.

Find the equation of a straight line passing through the point of intersection of x+2y+3=0 and 3x+4y+7=0 and perpenicular to the straight line x-y+0=0

Q. If the point $(a,a^2)$ lies inside the triangle formed by the lines $2x+3y-1=0,x+2y-1=0$ and $-8x+8y+2=0$ then

1. $-1<a<\frac{1}{2}$
2. $a>1$
3. $a>\frac{1}{3}$
4. $\frac{1}{3}<a<\frac{1}{2}$

Q. If $m_1$ and $m_2$ are the roots of the equation $x^2+(\sqrt{3}+2)x+(\sqrt{3}-1)=0,$ then the area of the triangle formed by the lines $y=m_{1}x,$ $y=m_{2}x$ and $y=2,$ is

1. $\sqrt{33}+\sqrt{11}$ sq. units
2. $\sqrt{33}-\sqrt{11}$ sq. units
3. $\sqrt{33}+\sqrt{7}$ sq. units
4. $\sqrt{33}-\sqrt{7}$ sq. units

Q.

Find the equation of the straight line which passes through the point intersection of the lines $3x-y=5$ and $x+3y=1$ and makes equal and positive intercepts on the axes.

Q.

ABCD is a square with side a(=9). Find the equation to the circle circumscribing the square if A is the origin

1. x² + y² + a(x + y) = 0

2. x² + y² = a(x - y)

3. x² + y² = a(x + y)

4. x² + y² + a(x - y) = 0

Q. The algebraic sum of distances of the line ax + by + 2 = 0 from (1, 2), (2, 1) and (3, 5) is zero and the lines bx – ay + 4 = 0 and 3x + 4y + 5 = 0 cut the co-ordinate axes at concyclic points then

1. 
2. area of the triangle formed by the line ax + by + 2 = 0 with coordinate axes is .
3. max {a, b} =
4. line ax + by + 3 = 0 always passes through the point (-1, 1)

Q. The sum of square of values of $c$ for which the equations
$2x+3y=3$
$(c+2)x+(c+4)y=(c+6)$
$(c+2{)}^{2}x+(c+4{)}^{2}y=(c+6{)}^2$ are consistent, is

Q. Let $S$ denotes the sum of all the values of $\lambda$ for which the system of equations
$(1+\lambda )x_1+x_2+x_3=1$
$x_1+(1+\lambda )x_2+x_3=\lambda$
$x_1+x_2+(1+\lambda )x_3={\lambda }^2$
is inconsistent. Then |S| is