Video Transcript
Find the length of the
perpendicular drawn from the point 𝐴 one, nine to the straight line negative
five 𝑥 plus 12𝑦 plus 13 equals zero.
So we’re going to answer this
question using the formula for calculating the distance between a point and a
straight line. So the formula is this. If I have the straight line
with equation 𝑎𝑥 plus 𝑏𝑦 plus 𝑐 is equal to zero and I have a point with
coordinates 𝑥 one, 𝑦 one. Then the perpendicular distance
between them, 𝑙, is given by the modulus of 𝑎𝑥 one plus 𝑏𝑦 one plus 𝑐, all
divided by the square root of 𝑎 squared plus 𝑏 squared. So what I need to do is
determine the values of 𝑎, 𝑏, 𝑐, 𝑥 one, and 𝑦 one and then substitute them
into the formula.
Let’s look at the straight line
first of all. I’m comparing it with 𝑎𝑥 plus
𝑏𝑦 plus 𝑐 is equal to zero. This shows me that 𝑎 is equal
to negative five, 𝑏 is equal to 12, and 𝑐 is equal to 13. Now let’s look at the point 𝐴,
which has coordinates one, nine. This tells me that 𝑥 one is
equal to one and 𝑦 one is equal to nine. So now I have all the values I
need. And it’s just a case of
substituting them into this formula for the distance 𝑙.
So we have that 𝑙 is equal to
negative five times one plus 12 times nine plus 13, the modulus of that
quantity. Then we’re going to divide it
by the square root of negative five squared plus 12 squared. This gives us the modulus of
negative five plus 108 plus 13 all divided by the square root of 25 plus
144. This gives the modulus of 116
over the square root of 169. Now as 116 is positive, then
its modulus is just its own value. So the numerator will be
116. And in the denominator, the
square root of 169 is 13 exactly.
So we have our answer to the
problem. The length of the perpendicular
between the point one, nine and the straight line negative five 𝑥 plus 12𝑦
plus 13 equals 0 is 116 over 13.
