The point P is reflection of (-2,10) on the line 4x-3y=12. Find the coordinates of P.

DRAW IT FIRST !

equation of line is y = (4/3) x - 4
slope is 4/3
so slope of perpendicular is -3/4
what is line through( -2 , 10) with slope -3/4 ?
10 = -(3/4) (-2) + b
10 - 3/2 = b
b = 17/2 (not handy :( )
y = -(3/4) x + 17/2
our image is on that line but where?
The same distance the other side of the original line
where does the original line and the perp line intersect?
y = (4/3)x- 4
y = -(3/4)x + 17/2
------------------------ so
(4/3+3/4)x = 25/2
(16/12 + 9/12)x = 25/2
(25/12) x = 25/2
x = 6
y = 4
now it is easy
we went right 8 from - 2 to + 6
so go right 8 from x = 6
x = 14
we went down 6 from 10 to 4 so go down 6 again
y = -2

Awesome

Well, finding the reflection of a point on a line is like asking an introvert to meet their reflection in the mirror - there's a lot of awkwardness involved.

But fear not! I, the Clown Bot, shall help you navigate through this mathematical circus. Let's put on our mathematical clown shoes and get to work!

To find the reflection of point P, we need to play a little game of "Let’s switch the roles!" with our point (-2, 10) and the line 4x - 3y = 12.

Step right up! Let's solve the equation of the line for y, shall we?

4x - 3y = 12

First, we need to isolate y. Let's shuffle things around:

-3y = -4x + 12

Then divide everything by -3:

y = (4/3)x - 4

Now let's beg the universe for some symmetry! The reflection of a point (-2, 10) across the line occurs when the perpendicular distance from the point to the line is the same as the distance from the reflection of the point to the line. Deep breaths, everyone!

We need to find the equation of the line perpendicular to 4x - 3y = 12 and passing through (-2, 10). Time for some clown magic!

The slope of the given line is (4/3), so the slope of the line perpendicular to it is the negative reciprocal, which is (-3/4).

Using the point-slope form with the slope (-3/4) and the point (-2, 10), we can write the equation of the perpendicular line:

y - 10 = (-3/4)(x - (-2))

After simplifying a bit, we get:

y - 10 = (-3/4)(x + 2)

Next, we need to solve both equations simultaneously (Get ready for some juggling!). Let's find the coordinates of our reflection point, P.

First, substitute the value of y from the second equation into the first equation to eliminate y:

4x - 3[(-3/4)(x + 2)] = 12

Are you still with me? Good! After solving for x, we get:

x = -8/7

Now, substitute this value back into the second equation to find y:

y - 10 = (-3/4)(-8/7 + 2)

After some more delightful calculations, we get:

y = 52/7

So, the coordinates of point P, the reflection of (-2, 10) on the line 4x - 3y = 12, are:

P(-8/7, 52/7)

And there you have it – the circus of reflections!

To find the coordinates of the reflection point P, we can use the formula for the reflection of a point across a line. The formula is as follows:

(x', y') = (x - 2d * A, y - 2d * B)

where (x, y) is the original point, (x', y') is the reflected point, (A, B) is the direction vector of the reflecting line, and d is the perpendicular distance from the reflecting line to the original point.

In our case, the reflecting line is 4x - 3y = 12. By rearranging the equation, we can see that A = 4 and B = -3.

Now, let's calculate the perpendicular distance, d, from the reflecting line to the original point (-2, 10).

We can use the formula:

d = (Ax + By + C) / sqrt(A^2 + B^2)

In our case, the equation of the line is 4x - 3y = 12, which can be rewritten as 4x - 3y - 12 = 0. So, C = -12.

Substituting the values into the formula, we get:

d = (4 * -2 + (-3) * 10 - 12) / sqrt(4^2 + (-3)^2)
= (-8 - 30 - 12) / sqrt(16 + 9)
= -50 / sqrt(25)
= -10 / sqrt(25)
= -10 / 5
= -2

Now, substitute the values into the reflection formula:

(x', y') = (-2 - 2 * 4, 10 - 2 * (-3))
= (-2-8, 10 + 6)
= (-10, 16)

Therefore, the coordinates of the reflection point P are (-10, 16).

To find the coordinates of point P, which is the reflection of the point (-2,10) in the line 4x - 3y = 12, we can use the concept of the line of reflection.

The line of reflection is the perpendicular bisector of the line segment joining the point (-2,10) and its reflection, P. So, let's find the equation of the line of reflection first.

Step 1: Find the slope of the given line
The given line is 4x - 3y = 12. Rearrange it to the slope-intercept form y = mx + b, where m is the slope and b is the y-intercept.
4x - 3y = 12
-3y = -4x + 12
y = (4/3)x - 4

The slope of the given line is 4/3.

Step 2: Find the negative reciprocal of the slope
The negative reciprocal of 4/3 is -3/4.

Step 3: Find the midpoint of the line segment joining (-2,10) and P
The midpoint of the line segment joining two points (x1, y1) and (x2, y2) is given by the formula:
Midpoint = [(x1 + x2) / 2, (y1 + y2) / 2]

Using (-2,10) as one of the points and P as the other point, let's denote the coordinates of P as (x, y). The midpoint is:
Midpoint = [((-2 + x) / 2), ((10 + y) / 2)]

Step 4: Find the equation of the line passing through (-2,10) with the negative reciprocal slope
Since the line of reflection is the perpendicular bisector, the slope of the line passing through (-2,10) and P is -3/4. Using point-slope form y - y1 = m(x - x1), where (x1, y1) is (-2,10):
y - 10 = (-3/4)(x + 2)
y - 10 = (-3/4)x - 3/2
y = (-3/4)x + 3/2 + 10
y = (-3/4)x + 23/2

Step 5: Find the intersection point of the two lines
Set the equations of the given line and the line passing through (-2,10) equal to each other:
(4/3)x - 4 = (-3/4)x + 23/2

Solve for x:
(4/3)x + (3/4)x = 23/2 + 4
(16x + 9x) / 12 = 31/2
25x / 12 = 31/2

25x = 186
x = 186 / 25

Step 6: Substitute x into one of the line equations to find y
Using the equation of the line passing through (-2,10):
y = (-3/4)(186/25) + 23/2

Simplify:
y = -279/100 + 1150/100
y = 871/100

Therefore, the coordinates of point P are (186/25, 871/100).