(i) Find the equation of tangent to curve `y=3x^(2) +4x +5` at (0,5)
(ii) Find the equation of tangent and normal to the curve `x^(2) +3xy+y^(2) =5` at point (1,1) on it
(iii) Find the equation of tangent and normal to the curve` x=(2at^(2))/(1+t^(2)) ,y=(2at^(2))/(1+t^(2))` at the point for which t`=(1)/(2)`
(iv) Find the equation of tangent to the curve `={underset(0" "x=0)(x^(2) sin 1//x)" "underset(x=0)(xne0)" at "(0,0)`

Answer

Let's solve the given problems step by step.

### (i) Find the equation of tangent to the curve \( y = 3x^2 + 4x + 5 \) at the point \( (0, 5) \).

1. **Differentiate the function**:
\[ \frac{dy}{dx} = \frac{d}{dx}(3x^2 + 4x + 5) = 6x + 4 \]

2. **Evaluate the derivative at \( x = 0 \)**:
\[ \frac{dy}{dx} \bigg|_{x=0} = 6(0) + 4 = 4 \]
The slope \( m = 4 \).

3. **Use the point-slope form of the equation of a line**:
\[ y - y_1 = m(x - x_1) \]
Substituting \( (x_1, y_1) = (0, 5) \):
\[ y - 5 = 4(x - 0) \]

4. **Rearranging gives the equation of the tangent**:
\[ y = 4x + 5 \]