Find the equation of the straight line

In summary, the conversation discusses a question with two parts involving finding coordinates and equations of lines. The participants use different approaches, such as Pythagorean theorem and gradients, to solve the problem and determine the coordinates of points B and D. They also discuss how to find the equation of a circle and its relationship with the equation of a line. Ultimately, they agree on the solution and conclude the conversation.
  • #1
chwala
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Homework Statement
See attached
Relevant Equations
straight line equations
Find the question here ( this one is a pretty easy question).

1640817062638.png


I have attempted this in the past using varied approach, this is in reference to part(b) of the question... i have previously used pythagoras theorem to finding co-ordinates of ##B## and ##D##...

Anyway find my current approach on this,

The gradient of the line ##AC=\frac{-1}{3}##, it follows that the gradient of line ##BD=3##, with mid-point of ##BD##=##(4,3)##.
For part (a),
The equation of line ##BD##, is given by
##y=mx+c##
##3=12+c##
##c=-9##,
Therefore, ##y=3x-9##

For part (b),
Let the co-ordinates of ##B##=##(x_1,y_1)##
##D##=##(x_2,y_2)##
then it follows that (using gradient),
$$\frac {3}{1}=\frac {3-y_2}{4-x_2}$$
$$⇒D(x_2,y_2)=(3,0)$$
Also,
$$\frac {3}{1}=\frac {y_1-3}{x_1-4}$$
$$⇒B(x_1,y_1)=(5,6)$$

Any other way for part (2) only. Cheers
 
Last edited:
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  • #2
AB is perpendicular to BC.

C and D are on the circle centered at (4,3) with radius ##\sqrt{10}##.
 
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Likes chwala
  • #3
$$(x-4)^2+(y-3)^2=10$$
$$(x-4)^2+(3x-12)^2=10$$
$$x^2-8x+15=0$$
##x_1=5,⇒y_1=15-9=6##
##x_2=3⇒y_2=9-9=0##

Bingo! anutta...
 
  • #4
The equation of the former of #2 is
[tex](x-7)(x-1)+(y-2)(y-4)=0[/tex]
which coincides with the equation of the circle.
 
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  • #5
anuttarasammyak said:
The equation of the former of #2 is
[tex](x-7)(x-1)+(y-2)(y-4)=0[/tex]
which coincides with the equation of the circle.
Agreed, i just used your approach in post ##3## unless you want me to show all steps. Its clear mate.:cool:
 

Related to Find the equation of the straight line

What is the definition of a straight line?

A straight line is a geometric figure that extends infinitely in both directions and has a constant slope.

How do you find the equation of a straight line?

To find the equation of a straight line, you need to know two points on the line. Then, you can use the slope formula, y = mx + b, where m is the slope and b is the y-intercept, to find the equation.

What is the slope of a straight line?

The slope of a straight line is the measure of its steepness. It is calculated by dividing the change in y-coordinates by the change in x-coordinates between two points on the line.

What is the y-intercept of a straight line?

The y-intercept of a straight line is the point where the line crosses the y-axis. It is represented by the value of b in the slope-intercept form of the equation, y = mx + b.

Can you find the equation of a straight line if you only know one point on the line?

No, you need at least two points on a line to determine its equation. If you only know one point, there are infinitely many lines that can pass through that point, each with a different equation.

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