Find the equation of the family of lines satisfying the following conditions :
`(i)` passing through the origin
`(ii)` parallel to the line `3x+4y+5=0`
`(iii)` having slope `5`
`(iv)` having `y`-intercept `4`.

Answer

To find the equation of the family of lines satisfying the given conditions, we will analyze each condition step by step.

### Step 1: Line Passing Through the Origin
A line that passes through the origin can be represented in the slope-intercept form as:
\[ y = mx \]
where \( m \) is the slope of the line.

**Hint:** Remember that a line passing through the origin has no constant term (y-intercept is 0).

### Step 2: Parallel to the Line \( 3x + 4y + 5 = 0 \)
To find the slope of the line given by the equation \( 3x + 4y + 5 = 0 \), we can rearrange it into slope-intercept form \( y = mx + c \):
\[ 4y = -3x - 5 \implies y = -\frac{3}{4}x - \frac{5}{4} \]
From this, we see that the slope \( m \) of the line is \( -\frac{3}{4} \).

Since we want lines that are parallel to this line, they will have the same slope. Thus, the equation of the family of lines parallel to this line can be written as:
\[ y = -\frac{3}{4}x + c \]
where \( c \) is a constant.

**Hint:** Parallel lines have the same slope.

### Step 3: Having Slope 5
Now, we need to find the equation of the line with a slope of 5. Using the slope-intercept form:
\[ y = mx + c \]
Substituting \( m = 5 \):
\[ y = 5x + c \]

**Hint:** The slope of the line is the coefficient of \( x \) in the equation.

### Step 4: Having y-intercept 4
The y-intercept of a line is the value of \( y \) when \( x = 0 \). If the y-intercept is 4, then when \( x = 0 \):
\[ y = 4 \]
Thus, we can write the equation of the line with a y-intercept of 4 as:
\[ y = 5x + 4 \]

**Hint:** The y-intercept is the constant term in the slope-intercept form when \( x = 0 \).

### Final Equation
The final equation of the line that satisfies all the conditions is:
\[ y = 5x + 4 \]