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Let E be an ellipse with equation (x/3)^2 + (y/4)^2 = 1. Find the equation of the tangent to E at the point P where x = √3 and y > 0, in the form ax + by = c, where a, b and c are rational.

In order to find the equation of the tangent to a curve at a point (x 1 , y 1 ), we use the equation y - y 1 = m(x - x 1 ), where m is the gradient of the curve at (x 1 , y 1 ). We already have x 1 = √3, so ...
BC
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Find the cube roots of unity.

Unity just means 1, so we need to find all z such that z 3 =1. However, although we can immediately see that z=1 is a solution, when dealing with complex numbers some solution may not lie on the real axis. T...
ET
9487 Views

Give the general solution to (d2y/dx2) - 2dy/dx -3y = 2sinx

Using the auxiliary equation t^2 - 2t - 3t = 0 t therefore is equal to 3 or -1. Using this value, a complementary function is derived. Y= Ae^(3x) + Be^(-x). Finally, to fully solve, a particular integral of ...
BY
10223 Views

Simplify (2x^3+8x^2+17x+18)/(x+2)

(x+2) | (2x 3 +8x 2 +17x+18) = 2x 2 +4x+9
SP
3259 Views

Expand (1+x)^3. Express (1+i)^3 in the form a+bi. Hence, or otherwise, verify that x = 1+i satisfies the equation: x^3+2*x-4i = 0.

First, we factor out one (x+1). (1+x)^3 = (x+1)^2(x+1)= Then, we expand using the formula (a+b)^2 = a^2 + 2ab +b^2: =(x^2+2x+1)(x+1)= Then we multiply: = x^3 + 2x^2 + x + x^2 + 2x + 1 We sum the terms with t...
AI
9051 Views