Right in equation of them on the passage to point BA is perpendicular to the law to give a “equation p(4,3), y=-x

Question

Right in equation of them on the passage to point BA is perpendicular to the law to give a “equation p(4,3), y=-x

Answer 1

Unfortunately, I'm not 100% sure with this one. Please let me rephrase...

There is a line with equation y = -x.

There is a point P(4,3) which is not on the line.

There is a second line, perpendicular the first, which passes through P. What is the equation of this line?

Is this what we want? I hope so!

I recommend you check out desmos.com/calculator as you can quickly enter the above line function, and try others to see what goes through P.

You can solve this problem using a tool like Desmos, or you can do it mathematically. However, I think it's good practice to do a bit of drawing so you can see what the problem "looks like". And, hopefully, you'd get a bit of credit for drawing a graph which looks like the problem definition.

The knowledge you need for this type of problem is the general equation for straight lines, y = mx + c.

You probably remember seeing something like this, and you know that m is the gradient, and c is the y intercept (where the line crosses the y axis).

The extra bit of knowledge you need is the rule about the gradients of perpendicular lines. This says that the product of the two gradients is -1. In maths-speak we say m1.m2 = -1

So... where do we start?

With the first line, y = -x.

How does this fit with y = mx + c?

y = -1x + 0 is the same as y = -x. so m = -1 and c = 0

Hopefully, you also remember that a positive gradient goes up and to the right, while a negative gradient goes down to the right. A gradient of 1 goes up one unit and across one unit.

So y = -x passes through (0,0) (-1, 1) (-2,2) (2,-2) etc and you can draw it.

Next mark the point P (4,3).

Finally, draw a line, perpendicular to the first, which goes through P.

Now look at your new line. Where does it cross the y axis?

Now for the derivation of the second equation... starting with the general y = mx + c.

If the first line (y = -x) has a gradient m1 of -1 then we know that -1.m2 = -1 hence m2 must be 1.

So now we know that the equation of the second line is y = x + c.

But we don't yet know c. But we do know this line goes through (4,3).

So we plug those values in to get 3 = 4 + c

which means c must be -1

Hence, the equation of this line is y = x - 1