Find the new coordinates of (2, 3) if the coordinate axes become x+y+1 = 0 and x-y+2=0?

There is nothing wrong in your method. May be it is little length. To the best of my knowledge there are two other methods - One is, using matrix factor for shifting origin and the other is using perpendicular distance form. I feel both these could be little shorter than the one what you have approached.

Anyway Answer is: (3√2, -√2/2)

Solution:

Method 1:

i) By solving the two line equations, their point of intersection with respect to conventional coordinate axes system = (-3/2, 1/2)

ii) The line x - y + 2 = 0 has a slope 1; so it makes 45 deg with the x-axis.

Hence the system is rotated by 45 deg in anti clockwise direction.

iii) When the origin is shifted from (0, 0) to (h, k) and rotation of the system in anticlockwise direction is u°,

the new coordinate is given by:

X = (x - h)*cos(u) + (y - k)*sin(u) and

Y = -(x - h)*cos(u) + (y - k)*cos(u)

iv) So here X = (2 + 3/2)*cos(45) + (3 - 1/2)*cos(45) = 3√2

Y = -(2 + 3/2)*cos(45) + (3 - 1/2)*cos(45) = -1/√2 = -√2/2

Thus in the new system it is: (3√2, - √2/2)

Method 2:

i) The line x - y + 2 = 0, has positive slope 1 and positive y intercept 2; so it lies above origin and making an angle of 45 deg with x-axis in anticlockwise direction; so let this be X-axis. Other line x + y + 1 = 0 is perpendicular with first one, making negative slope -1 and negative intercept -1; so it Y-axis in the new system.

ii) In the conventional system, the point (2, 3) lies below the first line and above the second line.

So with respect to new system XY, the point lies in 4th quadrant.

iii) The distance of the point (2, 3) in the conventional system will be the coordinates of (2, 3) in the new system.

Applying "Distance of a point (x₁, y₁), from the line ax + by + c= 0, is given by: |ax₁ + by₁ + c|/√(a² + b²)",

Distance of point (2, 3) from x - y + 2 = 0 is: (2 - 3 + 2)/√(1² + 1²) = 1/√2 = √2/2

As explained in (i) & (ii) above, this is y-coordinate and being in 4th, it is -√2/2

Distance of point (2, 3) from x + y + 1 = 0 is: 6/√2 = 3√2

Thus the coordinates in new system is: (3√2, - √2/2)

Note: This is my own original solution. Kindly be aware of one other member by name 'anjali', who has the habit of 100% copying the solution of others and present the same as of his/her own.

[If you are satisfied with the solution kindly acknowledge at least with a simple thanks. Here all of us are doing only a honorary service with no remuneration. So a simple acknowledgement may be a token of appreciation for the solvers, which ultimately bring more to provide solutions].

[It is edited for providing a solution as described in Method 2]

If you can get the right answer every time then there is nothing wrong with your method.

The co-ordinate axes are rotated by A=45 degrees clockwise as well as being translated. You have only taken account of the translation.

X = (x-h)cosA+(y-k)sinA

Y = (y-k)cosA-(x-h)sinA.

Here sinA=cosA=1/sqrt(2).