Maths AS Help!!!!!!!!``?

Question

FIND THE EQUATION OF THE LINE THROUGH THE GIVEN POINT WHICH IS PERPENDICULAR TO THE GIVEN LINE. YOUR FINAL ANSWER SHOULD NOT CONTAIN ANY FRACTIONS

(a) (2,3) y=4x + 3
(b) (5,-3) 2x=3

Could you please talk me through the answer that you get
Thanks, much appreciated.

Answer 1

Ok first you need to know the best way you can express lines:
A line of gradient 'm' which goes through the point (a,b) has equation:
y-b = m (x-a)
So you can get the gradient of the perpendicular line of (a) easily, because the perpendicular line is the negative reciprocal of the original, so in this case, -1/4.
So the line is:
y-3 = (-1/4) (x-2).
But, you're not supposed to have any fractions, so you can multiply the whole thing by 4:
4y-12 = - (x-2)
So: 4y-12 = 2-x.

(b) is different because it's not interms of y, but that means that the line must be vertical, so the perpendicular is horizontal.
if it's horizontal, all it's y values must be the same, so it can be written in the form:
y = something.

Since it goes through the point (5,-3), something must be -3.
So the equation for (b) is:
y = - 3.

Answer 2

The first thing to do is to work out the slope of the line. If y=mx+c then the slope of the line is m. The slope of a perpendicular line is -1/m. Therefore y= -x/m + d. You then need to fit that line to the point given. I.e. you put in the values you have for x and y, and find what d is equal to.

a) The original line is y=4x+3. The slope of the perpendicular line is therefore -1/4. So the equation is of the form y= -x/4+c. Put in (2,3), i.e. x=2, y=3 and you get 3= -2/4 + c. Rearrange that and c=3.5, so the equation of the line is y = -x/4 + 3.5 Multiply everything by 4 to lose fractions: 4y=14-x

b) The -1/m doesn't work for this one, as you don't have a y=mx+c. So what you want to do is draw the line. 2x=3 is the same as x=1.5. Draw this, and you get a vertical line. Perpendicular to that is a horizontal line, i.e. y is constant. If it's going to go through y= -3, then y must be -3 the whole time, and so the equation is y= -3, or y+3=0

Answer 3

There are 3 steps in general to this type of question:
1.) Find the slope of the new line.
2.) Find the offset of the new line in y-direction.
3.) Take the fractions out of the result.

So look at a)
1.) The slope of a perpendicular line to y=4x+3 is:
- 1 / 4. (The slopes m1 and m2 of two perpendicular lines
always coply to m1 = - 1 / m2).

2.) The new line y=-1/4 * x + b must go through the point (2,3)
so set x to 2 and you get:
y = -1/2 + b = 3 thus b is 3+1/2.
Thus the new line is:
y=-1/4 x + 3+1/2

3.) Now you want the result without fractions, so multipy the
equation with 4:
4 y= - x + 14

That is it.
In the case of b) you have a problem. 2x=3 is not a line - or
it could be considered a vertical line through x=3/2.
The result would then be a horizontal line through y=-3 (slope is zero).
The result is then
y=-3.

Does this help?

Answer 4

a) (y-f) =m(x-g)...........(1)

where m is gradient and ( g,f) is a point on the line

now we find the perpendicular to y=4x+3

the gradient of the perpendicuar line is the reciprocal to the gradient of the given line (m1*m2)=-1)

so, gradient of reciprocal line = -1/4

sub point(2,3) into (1), we have

(y-3)= -1/4(x-2) >>>>> 4(y-3)= (2-x)

>>>>> 4y+x-14 =0

therefore,the equation of the perpendicular to the line y=4x+3 passing through(2,3) is

4y+x-14 =0

b) the line 2x =3 is a vertical cutting the x-axis at x=3/2,therefore

the line perpendicular to this is a horizontal

at the point (5,-3) the line must pass through y = -3 -indeed,y will always be -3 since this line is horizontal and the point is x=5 when y= -3 -this line cuts the y-axis at y= -3

therefore,the equation of the perpendicular to the line 2x=3 passing through(5,-3) is

y= -3

Answer 5

okay..let me solve the (b) first,

since 2x=3 and x=3/2
it will be a straight line in the graph in which x=3/2 or 1.5

so the line which is perpendicular through the given point will be
y= -3 (draw a line through the given line from the given point 90 degrees)

for the question (a),

y=4x+3
the gradient will be 4
M1M2=-1
M2 = -1/4

using the given point (2,3),

y-3 = -1/4(x-2)
y-3 = -x/4 + 1/2
y = -x/4 + 7/2

the equation is y= -x/4 + 7/2
if u don't want the fraction, just multiply the equation by 4 and eventually u will get

4y = -x + 28

done..

Answer 6

Perpendicular lines have opposite reciprocal slope. For a, the line you've been given is in slope-intercept form, so it should be apparent to you that the slope is 4. The opposite reciprocal of 4 is -1/4. Now construct the line through (2,3) using point-slope form, y - k = m(x - h), where the line goes through the point (h,k) and has slope m. You already have your point and your slope. You may want to use simply algebra to put the equation into slope-intercept form. Since you need to get rid of all fractions, you'll need to multiply both sides of the equation by 4, and then you could rearrange that into standard form, Ax + By = D.

For b, the line is x = 3/2, a vertical line. That means that the perpendicular line is horizontal, with a constant y value. All you have to do is set y equal to the y-coordinate of the point you were given.

Answer 7

the slope of the line y = 4x+3 is 4
if line is perpendicular to this line slope is -1/m = -1/4
so equation of the line

y = -x/4 + c where c is a constant

or 4y = -x + 4c or -x + a where a is a constant

or x+ 4y = - a = b (-a can be put as b when b is constant)

now it passes through (2,3)

so 2 + 4.3 = b
b = 14
so the equation of line

x+ 4y = 14
(b)
for the second one 2x = 3 means x= 3/2
this is parallel to y axis

line perpendicular to this is perpendicular to x axis

y =c
passes through (5 ,-3) so it
is
y = -3

Answer 8

(a)
New gradient, m2 = -1/4
(because m1xm2=-1 for perpendicular lines)

y = mx+c
3 = (-1/4)(2) + c
3 = -1/2 + c
c = 3 1/2 = 7/2

So y = (-1/4)x + 7/2

Multiply both sides by 4 to remove the biggest denominator:
4y = -x + 14
x + 4y = 14

b)
As 2x=3 => x=3/2 is a vertical line, the perpendicular line must be horizontal, ie. gradient = 0.

Since it passes through (5,-3) and we know that it is horizontal, the equation must be y=-3.

Answer 9

The product of slopes of perpendicular lines=-1
y=4x+3
slope=4
Slope of the line we want shall be -4
It passes through the point 2,3
we use this data to get the constant c or intercept
y=mx+c
3=-4*2+c
c=3+8=11
the reqired equation is
y=-4x+11
You can work the second problem exactly the same way.

Answer 10

a) y =4x + 3
dy/dx = 4,

so gradient of perpendicular = -1/(dy/dx) = -1/4

use y-y1 = m(x-x1) to solve, m= -1/4. x1, y1 = (2,3)

so eqn is 4y=14-x


so to help you understand, to find the equation of the perpendicular at a given point you need to first find the gradient of your line, then, the gradient of the perpendicular is -1/gradient of line. then find the equation of the line. you try for b)