Coordinate geometry with given parameters.

In summary, the conversation is about a person seeking help with a problem in coordinate geometry. The problem involves finding the values of t for two different situations related to points A, B, and P, and the line l. The first part of the problem involves finding the values of t for which the length of AP is 5 units, and the second part involves finding the value of t for which OP is perpendicular to the line l. The conversation includes a discussion of using the distance formula and the point-slope formula to solve the problem. The person seeking help ultimately finds the correct answers for both parts of the problem.
  • #1
Aguy1
3
0
Hello,

Please forgive me if it's in the wrong sub-forum because don't know where to place it.

I need help solving this problem it's chapter 1 and in our class we are already in chapter 5 so I might sound like a fool asking the teacher about it, I was revising and decided to do some questions then I stumbled on this question and got stuck.

Q.The coordinates of the points A and B are (2,3) and (4,-3) respectively,coordinates of P is (2+2t,3-t)

1.Find the values of t such that the length of AP is 5 units.

2. Find the value of t such that OP is perpendicular to l (where O is the origin). Hence, find the length of the perpendicular from O to l.

Just in case here are the answers 1. sqrt5 or -sqrt5 2. t= (-1/5,8/5 sqrt5)

I know in the first I should use the distance formula but I'm not getting the right answer.

I'm sorry if I sound too demanding.
 
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  • #2
Re: Coordinate geometry.

Hello aguy1,
lets say we got \(\displaystyle A=(x,y)\) and \(\displaystyle B=(p,g)\) Then \(\displaystyle AB=(p-x,g-y)\) and the length of A is \(\displaystyle |A|=\sqrt{x^2+y^2}\) does this help you?

Regards,
\(\displaystyle |\pi\rangle\)
 
  • #3
Re: Coordinate geometry.

Petrus said:
Hello aguy1,
lets say we got \(\displaystyle A=(x,y)\) and \(\displaystyle B=(p,g)\) Then \(\displaystyle AB=(p-x,g-y)\) and the length of A is \(\displaystyle |A|=\sqrt{x^2+y^2}\) does this help you?

Regards,
\(\displaystyle |\pi\rangle\)

Thanks a lot, I got the answer for the first one :).

what about second question? I'm guessing O coordinates is (0,0) since it's the origin and I know the gradient should be perpendicular m1*m1=-1, so which equation should I use now?

Again thank you.
 
  • #4
Re: Coordinate geometry.

You haven't told us what "l" is...

if l (I assume this is a lower case L) is the line traced out by the point P as t varies:

First, we need to find the slope of this line. we can write:

$l = (2,3) + t(2,-1)$

which makes it clear the slope of l is -1/2. Thus the perpendicular line has slope 2. Now solve:

$2 + 2t = 2(3 - t)$

(that is: find the point (x,2x) (the point on the line through the origin y = 2x) that lies on l).
 
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  • #5
Re: Coordinate geometry.

Deveno said:
You haven't told us what "l" is...

Oh I'm very sorry, no coordinates of l just the line equation which is x+2y=8.

I hope you can forgive my foolishness.

It's a four part question but I finished both and the other two (which are the above, one which I just solved)

just for more info here are the two questions which I already solved.

(a) Find the equation of the line l through the point A(2,3) with gradient -1/2 (that's how I got l equation x+2y=8)

(b) Show that the point P with coordinates (2+2t, 3-t) will always lie on l whatever the value of t.
 
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  • #6
Re: Coordinate geometry.

So my guess was correct, then, as:

$(2,3) + t(2,-1)$ passes through the points (2,3) (t = 0), and (4,2) (t = 1), so the slope is:

$\frac{2 - 3}{4 -2} = -\frac{1}{2}$.

Using the point-slope formula:

$y - y_1 = m(x - x_1)$ and the point (4,2)

We get:

$y - 2 = -\frac{1}{2}(x - 4)$
$y - 2 = \frac{-x}{2} + 2$
$y - 4 = \frac{-x}{2}$
$2y - 8 = -x$
$x + 2y = 8$

so it is the same line.
 

Related to Coordinate geometry with given parameters.

1. What is coordinate geometry?

Coordinate geometry is a branch of mathematics that deals with the study of geometric figures using a coordinate system. It involves analyzing the position of points on a plane or in space using their coordinates, which are typically represented by ordered pairs or triplets.

2. What are the basic concepts of coordinate geometry?

The basic concepts of coordinate geometry include the use of a coordinate plane, the Cartesian coordinate system, and the distance formula. These concepts are used to represent and study points, lines, and shapes on a plane or in space.

3. How are points represented in coordinate geometry?

Points in coordinate geometry are represented by ordered pairs or triplets of numbers, depending on the number of dimensions. For example, in a two-dimensional coordinate plane, a point (x,y) represents the x and y coordinates of the point. In a three-dimensional space, a point (x,y,z) represents the x, y, and z coordinates of the point.

4. What is the importance of parameters in coordinate geometry?

Parameters in coordinate geometry refer to the values that are used to represent geometric figures or equations. These values can be adjusted to change the position, size, or shape of a figure, allowing for the study and analysis of different configurations. Parameters are also important in calculating equations and solving problems in coordinate geometry.

5. How is coordinate geometry used in real life?

Coordinate geometry has numerous applications in real life, such as in navigation, map-making, architecture, and computer graphics. It is also used in fields such as physics, engineering, and economics to model and analyze various systems and phenomena. Additionally, coordinate geometry is the basis for many advanced mathematical concepts and theories.

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