Cartesian Coordinate System

 What will be the area of the rhombus $ax \pm by \pm c = 0$?

How many points on $x + y = 4$ are there that lie at a unit distance from the line $4x + 3y = 10$?

If $p$ is the length of the perpendicular from the origin to the line $\frac{x}{a}+\frac{y}{b}=1$, then which of the following is true?

Find the value of $k$ if the straight line $2x + 3y + 4 + k (6x - y + 12) = 0$ is perpendicular to the line $7x + 5y = 4$.

Two sides of a square lie on the lines $x + y = 1$ and $x + y + 2 = 0$. What is its area?

$P (3, 1), Q (6, 5)$ and $R (x, y)$ are three points such that the angle $PRQ$ is a right angle and the area of $∆PRQ$ is $7$. The number of such points $R$ that are possible is

The equations of two equal sides $AB$ and $AC$ of an isosceles triangle $ABC$ are $x + y = 5$ and $7x - y = 3$ respectively. What will be the length of the intercept cut by the side $BC$ on the y-axis?

If the coordinates of the points $A, B, C$ and $D$ are $(6, 3), (-3, -5), (4, -2)$ and $(a, 3a)$ respectively and if the ratio of the area of triangles $ABC$ and $DBC$ is $2 : 1$, then the value of $a$ is

If $P$ and $Q$ are two points on the line $3x + 4y = -15$, such that $OP=OQ = 9$ units, the area of the triangle $POQ$ will be

If the points $(a, 0), (0, b)$ and $(1, 1)$ are collinear then which of the following is true?

The extremities of a diagonal of a parallelogram are the points $(3, -4)$ and $(-6, 5)$. If the third vertex is the point $(-2, 1)$, the coordinate of the fourth vertex is

$PQR$ is an isosceles triangle. If the coordinates of the base are  $Q (1, 3)$ and $R (-2, 7)$, then the coordinates of the vertex $P$ can be

The point of intersection of the lines $\frac{x}{a}+\frac{y}{b}=1$ and $\frac{x}{b}+\frac{y}{a}=1$ lies on the line.

What are the points on $x$ -axis whose perpendicular distance from the straight line $\frac{x}{a}+\frac{y}{b}=1$ is $a?$

The area of triangle, formed by the points $(p, 2-2p), (1-p, 2p)$ and $(-4-p, 6-2p)$ is $70$ units. How many integral values of $p$ are possible?

One side of the rectangle lies along the$4x+7y+5=0$. Two of its vertices are$(-3,1) \mathrm{and} (1,1).$ Find the equation of other three sides.

The area of a triangle is $5$ square units. Two of its vertices are $(2, 1)$ and $(3, -2)$. The third vertex lies on $y = x + 3$. What will be the third vertex?

A line passing through the points $(\alpha ,2\alpha )$ and $(-2,3)$ is perpendicular to the line $4x+3y-7=0$, find the value of $\alpha$.

What will be the value of $p$ if the equation of straight line $2x + 5y = 4$ gets changed to $2x + 5y = p$ after shifting the origin at $(3, 3)$?

What will be the new equation of a straight line $3x + 4y = 6$ if the origin gets shifted to $(3, – 4)$?