Find the angle between the lines 3x-4y +6 = 0 and 4y +6 = 0

Find the distance of the point (-2,1) from the line 3x-4y +6=0.

Find Find the inflection points of f(x) and determine the concavity around them. d dy/dx for the followings

Check orthogonality for the circles x²+y²+4x+6y+9=0 and x²+y²-6x-8y+15=0.

Find the radical axis of the circles x² + y²+ 4x + 6y + 9 = 0 and x² + y² - 6x - 8y + 15=0.

Find the chord of contact from (7,2) to the circle x²+y²=9.

Compute the radius and center of curvature for f(x) = x³-4x+15.

Find Find the inflection points of f(x) and determine the concavity around them. d dy/dx for the followings

Compute the radius and center of curvature for f(x) = x³-4x+15.

Evaluate the following limit.

Find an equation for the tangent line to the curve y = 2/x at the point (2,1).

Show that Mean Value Theorem holds for f(x) = x²

Show that Rolle's Theorem holds for f(x) = x²-4x+6.

Expand the following term using Leibniz's Theorem

Show that the function

is continuous at the point x = 1 but f'(1) does not exist.

Find the volume of the largest possible right circular cylinder that can be inscribed in a sphere of radius a. 

Find the first four terms in the expansion of ln sin(x + h) in ascending powers of h. Hence find the value of ln sin 31° to four places of decimals, given ln 2 = 0.69315.

Find dy/dx (any two) if:

(i) x = at², y = 2at,   (ii) y = (sinx) ^cosx  and (iii) y = tan^-1 (sec x + tan x)

If y = cos(m sin^-1 x) then prove that (1 - x²)y_n+2 - (2n + 1)xy_n+1 + (m² - n²)yⁿ = 0.

If, 

be finite, find the value of a and also the limit.

If the tangent to the curve √x + √y = √a at any point on it cuts the two axes at P and Q, show that OP + OQ = a, O being the origin.

Find the intervals on which the function f(x) = 2x³-3x² - 36x + 7 is increasing or decreasing. Also find the maxima and minima of f(x).

Find the equation of circle of curvature of the curve y = sinx at the point (π/2, 1).