Coordinate Geometry

Find the equation of a line with slope $m = -1$ and passing through the point $(0, 5)$.

A line passes through $(0,0)$ and $(-3, -3)$. What is the angle of inclination of this line with the positive direction of the x-axis?

What is the gradient (slope) of the line $2x + 5y - 10 = 0$?

The general equation of a straight line is $3x - 4y + 12 = 0$. What is the y-intercept of this line?

Find the equation of the line passing through the intersection of $2x + y = 5$ and $x + 3y = 8$ and the point $(1, 1)$.

Find the angle between the lines $y = x$ and $y = \sqrt{3}x$.

Convert the equation $\sqrt{3}x + y + 2 = 0$ into normal form $x \cos \alpha + y \sin \alpha = p$. What is the value of $p$?

Find the coordinates of the orthocentre of the triangle whose sides are $x + y = 1$, $x - y = 1$, and $y = 0$.

Find the ratio in which the line $3x + 4y = 7$ divides the distance between the lines $3x + 4y = 5$ and $3x + 4y = 10$.

A person standing at a junction of two straight paths represented by $2x - 3y + 4 = 0$ and $3x + 4y - 5 = 0$ wants to reach the path $6x - 7y + 8 = 0$ in the least time. Find the equation of the path he should follow.

Find the perpendicular distance from the point $P(3, 4)$ to the line $3x - 4y + 12 = 0$.

Find the perpendicular distance from the point $(2, -1)$ to the line $x - y = 5$.

Find the distance of the line $4x - y = 0$ from the point $(4, 1)$.

The distance of a point $(x_1, y_1)$ from the line $Ax + By + C = 0$ is zero if:

The distance between lines $L_1$ and $L_2$ is $d$. If $L_1$ is $2x + 3y = 6$ and $L_2$ is $2x + 3y = k$, and $d = \sqrt{13}$, find $|k-6|$.

Which point on the y-axis is at a distance of $5$ units from the line $4x - 3y - 12 = 0$?

Find the distance of the point $(0, 0)$ from the line joining $(a\cos\alpha, a\sin\alpha)$ and $(a\cos\beta, a\sin\beta)$.

A light ray is emitted from point $(1, 2)$ and reflects off a mirror placed along the $x$-axis. The reflected ray passes through the point $(5, 3)$. Find the perpendicular distance of the point $(5, 3)$ from the line representing the incident ray path (extended).

Find the distance between the point of intersection of the lines $x - y + 1 = 0$ and $2x + y - 7 = 0$ and the line $3x + 4y + 5 = 0$.

A moving particle $P(x, y)$ is such that its distance from the line $3x - 4y + 1 = 0$ is always $2$ units. Which of the following equations represents one such possible path (locus) of the particle?

What is the focal length (distance from vertex to focus) of the parabola $y^2 = 20x$?

Find the length of the minor axis of the ellipse $9x^2 + 4y^2 = 36$.

Identify the type of conic section represented by the equation $x^2 + 2y^2 = 8$.

Find the coordinates of the center of the circle $x^2 + y^2 - 4x + 6y - 12 = 0$.

Find the point of intersection of the asymptotes of the hyperbola $\frac{(x-2)^2}{9} - \frac{(y+3)^2}{16} = 1$.

Determine the length of the latus rectum of the ellipse $4x^2 + 9y^2 = 36$.

Identify the equation of the line that is the directrix of the parabola $x^2 = -12y$.

An arch is in the form of a semi-ellipse. It is $8$ meters wide and $2$ meters high at the center. Find the height of the arch at a point $1.5$ meters from one end.

A line $y = mx + 1$ is tangent to the parabola $y^2 = 4x$. Find the value of $m$ and the coordinates of the point of tangency.

A circle is tangent to the $y$-axis at $(0, 3)$ and makes an intercept of $8$ units on the positive $x$-axis. Find the radius of the circle.