Analytic Geometry

The inclination of the line determined by the points (4, 0) and \((5, \sqrt{3})\) is

How many petals are there in the rose curve \(r = 3\cos 5\theta\)?

How far is the directrix of the parabola \((x - 4)^2 = -8(y - 2)\) from the x-axis?

The equations of two lines are \(3y - 2x = 6\) and \(3x + ky = -7\). For what value of k will the two lines be parallel?

The vertex of the parabola \(y = (x - 1)^2 + 2\) is ______.

If the parabola \(y = x^2 + C\) is tangent to the line \(y = 4x + 3\), find the value of C.

A parabola having its axis along the x-axis passes through (−3, 6). Compute the length of the latus rectum if the vertex is at the origin.

Points (6, −2) and (a, 6) are on a line with a slope of 4/3. What is the value of a?

If the line \(kx + 3y + 8 = 0\) has a slope of 2/3, determine k.

What is the length of the latus rectum of the parabola \(x^2 = -16y\)?

What is the equation of the normal to the curve \(x^2 + y^2 = 25\) at (4, 3)?

Determine the eccentricity of the hyperbola \(xy = 8\).

If the graph of \(y = (x - 2)^2 - 3\) is translated 5 units up and 2 units to the right, then the equation of the graph obtained is

A line with equation \(y = mx + b\) passes through (−1/3, −6) and (2, 1). Find the value of m.

The eccentricity of the hyperbola having the rectangular equation \(3x^2 - 4y^2 - 24x + 16y + 20 = 0\) is

Find the slope of the line through the points (−2, 5) and (7, 1).

For what value of k will the line \(kx + 5y = 2k\) have a y-intercept of 4?

Find the point along the line x = y = z that is equidistant from (3, 0, 5) and (1, −1, 4).

An equilateral triangle is inscribed in the parabola \(x^2 = 8y\) such that one of its vertices is at the origin. Find the length of the side of the triangle.

Find the slope of the line passing through the pair of points (−2, 0) and (3, 1).