1). Given: Point ( 6 , 16 ) and equation of line is 2x + 5y = 4

First we find the slope of given line by writing it in slope intercept form.

So, the slope is -2/5

We also knows that product of slopes of perpendicular lines equal to -1

let slope of required line is m the,

Now, The equation of required line ,

Therefore, Option C is correct.

2). Given: equation of line is 3x + 6y = 9

we find the slope of given line by writing it in slope intercept form.

So, the slope is -3/6 = -1/2

We also knows that Slope of parallel lines are same.

Therefore, Option A is correct.

3). Given: equation of line is 3x + 4y = 7

First we find the slope of given line by writing it in slope intercept form.

So, the slope is -3/4

We also knows that product of slopes of perpendicular lines equal to -1

let slope of required line is m the,

Now, we check slope of each option.

Option A- 4x + 3y = 3

y = -4/3x + 3/3

⇒ Slope = -4/3

Option B- 4x - 3y = 3

y = 4/3x - 3/3

⇒ Slope = 4/3

Therefore, Option B is correct.

4). Given: Point ( -2 , 4 ) and equation of line is 2x + y = 4

First we find the slope of given line by writing it in slope intercept form.

So, the slope is -2

We also knows that Slope of parallel lines are equal.

Thus, slope of required line is -2

Now, The equation of required line ,

Therefore, Option D is correct.

5). Given: equations of line are 3x - y = -7 and 6x - 2y = 1

First we find the slope of given lines by writing it in slope intercept form.

3x - y = -7

y = 3x + 7

So, Slope of 1st line = 3

So, the slope of 2nd line is 6/2 = 3

Since slope of both lines are equal. given lines are parallel.

Therefore, Option A is correct.