Straight lines practice · Easy

The slopes of the line which passes through the origin, and the mid-point of the line segment joining the points. $P(3,-4)$ and $Q(-5,-2)$ is

Given a family of lines $a\left( {2x + y + 4} \right) + b\left( {x - 2y - 3} \right) = 0$, the number of lines belonging to the family at a distance $\sqrt {10}$ from $P\left( {2,\; - 3} \right)$ is

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 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 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

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

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?

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

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

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

The coordinates of the incentre of the triangle having sides

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

and $y - 15 = 0$ are

Normal form of line $x+y+1=0$ is

Consider the following statements.

I. The equation of a straight line making intercepts $a$ and $b$ on $x$ and $y$-axis respectively is given by $\frac{x}{a}+\frac{y}{b}=1$

II. If $a x+b y+c_1=0$ and $a x+b y+c_2=0$ be two parallel lines, then distance between two parallel lines, $d=\left|\frac{c_1-c_2}{\sqrt{a^2+b^2}}\right|$.

Choose the correct option.

A line has slope m and y-intercept 4. The distance between the origin and the line is equal to

If $\left( { - 2,6} \right)$ is the image of the point (4,2) with respect to the line L=0, then L=

If the three distinct lines $x+2 a y+a=0, x+3 b y+b=0$ and $x+4 a y+a=0$ are concurrent, then the point $(a, b)$ lies on a:

If $A$ and $B$ are two points having coordinates (3, 4) and $\left( {5, - 2} \right)$ respectively and $P$ is a point such that $PA = PB$ and area of triangle $PAB = 10$ sq unit, then the coordinates of $P$ are

The image of the point (3,5) in the line $x - y + 1 = 0$, lies on