Question Video: Determining the Shortest Distance between a Line and a Point Mathematics • First Year of Secondary School

Video Transcript

Find the shortest distance between the line 𝑦 equals a half 𝑥 minus two and the point 𝐴: nine, negative 10.

Well, if we think about what this question is asking and if we’re trying to find the shortest distance between a line and a point, then this is gonna be the perpendicular distance. Because if we take a look at the little sketch I’ve drawn here, if I wanted to get from the point where I’ve drawn here, which is the x point, and I wanted to get to the line, then the shortest root would be directly to it, which is gonna be perpendicular.

So what we actually have is a formula that allows us to find the perpendicular distance for a point at a line. Well, to use this, we have to have the equation of our line in the form 𝑎𝑥 plus 𝑏𝑦 plus 𝑐 equals zero. And we also have a point which is 𝑥 sub one, 𝑦 sub one. And then to find out our formula, what we have is our 𝐿, which is our perpendicular distance — well, in this case, the shortest distance — is equal to the modulus or absolute value of 𝑎𝑥 sub one plus 𝑏𝑦 sub one plus 𝑐 all over the square root of 𝑎 squared plus 𝑏 squared.

So, first of all, what we have is 𝑦 is equal to a half 𝑥 minus two. So then if we subtract 𝑦 from each side of our equation, what we’re gonna get is a half 𝑥 minus 𝑦 minus two equals zero. I’ve just flipped it here so that we have the expression on the left-hand side. So then what we do is we identify our 𝑎, 𝑏, and 𝑐. Well, in this case, the coefficient of 𝑥 is a half, the coefficient of 𝑦 is negative one, and then we’ve got 𝑐 is equal to negative two. And then what we have is that our point 𝐴, so nine, negative 10, is gonna give us our 𝑥 sub one of nine and our 𝑦 sub one of negative 10.

So, great, we now have all the parts we need. Let’s substitute them into our equation to find our perpendicular distance or shortest distance between our line and the point. So when we substitute into our formula, we’re gonna get 𝐿 is equal to the modulus or absolute value of a half multiplied by nine plus negative one multiplied by negative 10 minus two. And this is all over the square root of a half squared plus one squared. So, therefore, 𝐿 is gonna be equal to the modulus or absolute value of nine over two plus 10 minus two all over the square root of a quarter plus one.

So now, what we’re gonna do is tidy this up. So we can do that by, for the numerator, converting the other two values to halves. So, for instance, 10 is the same as 20 over two or twenty-halves, and two is the same as four over two or four-halves. And then, on the denominator, we’ve got one where one is the same as four-quarters. So what this is gonna give us is that 𝐿 is equal to the modulus or absolute value of 25 over two over the square root of five over four.

Well, we can remove the modulus or absolute value because this would just make our answer to be positive on the numerator when in fact it’s already a positive value. So now, we’ve got what we’ve got at this stage. So now, let’s simplify. Well, to help us do this, we have a rule. And that is one of our surd rules. And if we have root 𝑎 over 𝑏, it’s the same as root 𝑎 over root 𝑏. So now, we’ve got 25 over two divided by root five over two. And that’s because root four is equal to two. Well, now, what we can do is use one of our rules for calculating using fractions. And that is if we divide by a fraction, it’s the same as multiplying by reciprocal of the fraction. And the reciprocal of the fraction is when we swap the numerator and denominator.

So what we’ve got here is 25 over two multiplied by two over root five. So what we could do now is cross cancel. And that’s because if we divide the numerator and denominator both by two, we’ve got two values here that would just be canceled or be leaving us with one. So we’ve got 25 multiplied by one over one multiplied by root five, which is gonna give us 25 over root five. Now, we’ve got 25 over root five. However, what we want to do here is rationalize this denominator because we don’t want to have a surd as our denominator. So if we rationalize this denominator, what we’re gonna do is do 25 over root five multiplied by root five over root five. And we do this because if we’re gonna multiply root five by root five, then this is just gonna give us five.

So as we said, we can use another surd rule here. And if we have root five multiplied by root five, this is gonna be five because root 𝑎 multiplied by root 𝑎 is just 𝑎. So we’re left with 𝐿 is equal to 25 root five over five, which gonna leave us with our final answer which tells us that the shortest distance between the line 𝑦 equals a half 𝑥 minus two and the point 𝐴 nine, negative 10 is five root five.