Let's do this question, find the coordinates of the foot of the perpendicular from three comma five
to the nine X plus two is equal to eight.
All right, what does it mean?
We have a different line that this line B, the given line X plus two is equal to.
Now we have a point three comma five.
Now we need to draw a perpendicular from three five to this line.
But this angle over here is 90 degrees.
And we need to find this point, OK, to pause the video, give it a try, and then let's do it together.
All right.
We are back.
Let's start finding the equation of this line over here.
That is the line that is perpendicular to the given line.
Now, over here, the given line is two way plus X, minus eight is equal to zero.
Right now, we can write the equation of this line that is perpendicular to the given line as Y minus
two explosive is equal to zero.
Right.
All we did is we interchange the coefficients and we change the sign over here.
Now we know that this line, that is this line over here passes through three comma five.
Right.
So let's substitute three comma five into this equation.
All right, so that gives us five minus two in the three plus CS equal to zero, right.
And hence C is equal to one.
All right.
So the equation of this line over here is Y minus two, X plus one is equal to zero.
Right now, we know that this line and the initial line with which we started off intersect at this
point, which is the point that we are trying to find.
So let's try to solve between this equation and this equation for that.
Let's multiply this equation with two.
So I get to a minus four, X plus two is equal to zero.
Right.
And the given equation is to Y plus X minus eight is equal to zero.
So let me subtract these two equations or two and minus two is equal to zero minus four X minus gives
me minus five X and two minus minus eight gives me 10 to minus five X plus ten is equal to zero or X
is equal to two.
All right.
Now substituted into the given equation and you get Y is equal to three.
Right.
Therefore the point of intersection is to come up three.
Now let's discuss one more method in which we can solve this question.
Let me make some space over here now.
We will see this other method.
So this is a given nine X plus two is equal to eight and we have three come on five, which is this
point right now.
We are dropping a perpendicular and we need to find the coordinates of a comma.
Maybe that's the question right now.
What would be the slope of this line?
We know how to find the slope.
If you have two points, the slope of this line will be B minus five, divided by A minus three.
So this is the slope of this line and the slope of the given line over here is minus one by two.
Right.
Therefore, this slope over here has to be equal to two.
Right, because these are perpendicular.
So when you multiply them, you should get minus one.
Now let's solve this.
So you get B minus five is equal to A minus six.
That gives to A minus B is equal to one, right?
So we have this equation now as A and B is on this line over here, we can substitute it into the given
line.
Also right now on substitution, you will get A plus two.
B is equal to eight.
Right.
So, again, you have two variables and you have two equations.
You can solve it by multiplying this equation throughout with two and you will get B, and so by substituting
back, you will get A and you will get the same answer.
That is on solving.
You'll get the point about two, comma three.
And that is the answer to this question.