Straight lines practice · Mixed difficulty

For $a>0, b>0, c>0$, the distance between $(1,1)$ and the point of intersection of the lines $a x+b y+c=0$ and $b x+a y+c=0$ is less than $2 \sqrt{2}$. Then,

The measure of the angle between the lines $x \cos \beta+y \sin \beta=\frac{5 \pi}{2}$ and $x \cos \alpha+y \sin \alpha=\frac{\pi}{2}$ is

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

The straight line whose sum of the intercepts on the axes is equal to half of the product of the intercepts, passes through the points

If $P=(1,0), Q=(-1,0)$ and $R=(2,0)$ are three given points, then locus of the point $S$ satisfying the relation $S Q^2+S R^2=2 S P^2$, is

The value of $\lambda$ for which the lines joining the point of intersection of curves $C_1$ and $C_2$ to the origin are equally inclined to the $X$-axis, where $C_1: \lambda x^2+3 y^2-2 \lambda x y+9 x=0$ and $C_2: 3 x^2-4 y^2+8 x y-3 x=0$, is

If the quadrilateral formed by the lines $a x+b y+c=0, a^{\prime} x+b^{\prime} y+c=0, a x+b y+c^{\prime}=0$,
$a^{\prime} x+b^{\prime} y+c^{\prime}=0$ have perpendicular diagonals, then

The image of the point (4,-3) with respect to the line y=x is

If $a, b$ and $c$ are any three terms of an AP, then the $a x+b y+c=0$

The equation of the straight lines passing through the point (4, 3) and making intercepts on the coordinate axes whose sum is -1, is

The equation of the line $A B$ is $y=x$. If $A$ and $B$ lie on
the same side of the line mirror $2 x-y=1$, then the equation of the
image of $A B$, is

If $x^2-y^2+2 h x y+2 g x+2 f y+c=0$ is the locus of a point, which moves such that it is always equidistant from the lines $x+2 y+7=0$ and $2 x-y+8=0$, then the value of $g+c+h-f$ equals

The centroid of the triangle formed by the pair of straight lines $12{x^2} - 20xy + 7{y^2} = 0$ and the line $2x - 3y + 4 = 0$ is

If $a \ne b \ne c,$ then the equations $\left( {b - c} \right)x + \left( {c - a} \right)y + \left( {a - b} \right) = 0$ and, $\left( {{b^3} - {c^3}} \right)x + \left( {{c^3} - {a^3}} \right)y + \left( {{a^3} - {b^3}} \right) = 0$
will represent the same line, if

Area of the parallelogram formed by the lines $y=m x, y=m x+1, y=n x$ and $y=n x+1$ equals

If the three points $\left( {3q,0} \right),\left( {0,3p} \right)$ and $\left( {1,1} \right)$ are collinear then which one is true?

The acute angle formed between the lines joining the origin to the points of intersection of the curves $x^2+y^2-2 x-1=0$ and $x+y=1$, is

A rectangle $A B C D, A \equiv(0,0), B \equiv(4,0), C \equiv(4,2)$ and $D \equiv(0,2)$ undergoes the following transformations successively:

(i) $f_1(x, y) \rightarrow(y, x)$

(ii) $f_2(x, y) \rightarrow(x+3 y, y)$

(iii) $f_3(x, y) \rightarrow\left(\frac{x-y}{2}, \frac{x+y}{2}\right)$

The final figure will be

Given four lines whose equations are $x + 2y - 3 = 0,\;2x + 3y - 4 = 0,3x + 4y - 7 = 0$ and $4x + 5y - 6 = 0,$ then the lines are

Coordinate of the foot of the perpendicular drawn from (0, 0) to the line joining $(a\mathrm{\;cos\alpha },\mathrm{\;}a\sin \mathrm{\alpha })$ and $(a\cos \mathrm{\beta },\mathrm{\;}a\sin \mathrm{\beta })\;$ are

The equation of the bisector of the acute angle between the lines $2 x-y+4=0$ and $x-2 y=1$, is

Let $A$ be the point of intersection of the lines $3 x+2 y=14,5 x-y=6$ and $B$ be the point of intersection of the lines $4 x+3 y=8,6 x+y=5$. The distance of the point $P(5,-2)$ from the line $A B$ is

The angle between the straight line $x - y\sqrt 3 = 5$ and $\sqrt 3 x + y = 7$ is

The lines $ax + by + c = 0,\;bx + cy + a = 0$ and $cx + ay + b = 0$ $\left( {a \ne b \ne c} \right)$ are concurrent, if

Coordinates of the foot of the perpendicular drawn from (0, 0) to the line joining $(a\cos \mathrm{\alpha },a\sin \mathrm{\alpha }$) and ($a\cos \mathrm{\beta },a\sin \mathrm{\beta }$) are

Consider three points

$P = \left( { - \sin \left( {\mathrm{\beta } - \mathrm{\alpha }} \right),\mathrm{\;} - \cos \mathrm{\beta }} \right)$

${\cal Q} = \left( {\cos \left( {\mathrm{\beta } - \mathrm{\alpha }} \right),\mathrm{\;}\sin \mathrm{\beta }} \right)$

And $R = \left( {\cos \left( {\mathrm{\beta } - \mathrm{\alpha } + \mathrm{\theta }} \right),\mathrm{\;}\sin (\mathrm{\beta } - \mathrm{\theta }} \right))$,

Where $0 < \alpha ,\;\beta ,\;\theta < \frac{\pi }{4}$. Then,

Consider the following statements.

I. Centroid of a triangle is a point where angle bisectors meet.

II. If value of area after calculations is negative then we take its negative value.

Choose the correct option.

A triangle ABC, right angled at A, has points A and B as (2,3) and (0,-1) respectively. If BC=5 units, then the point C, is

Family of lines $\lambda x+3 y-6=0(\lambda$ is a real parameter $)$ intersect the lines $x-2 y+3=0$ and $x-y+1=0$ in $P$ and $Q$, then locus of the middle point of $P Q$ is

The coordinates of the incentre of the triangle having sides

$3x - 4y = 0,\;5x + 12y = 0$

and $y - 15 = 0$ are