Video Transcript
Given that ๐ถ๐ท is a diameter of the circle ๐, and the coordinates of the points ๐ and ๐ท are negative 11 over two, negative one and negative seven, seven respectively, determine the equation of the tangent to the circle at the point ๐ถ.
So, to help us understand whatโs going on in this question, Iโve drawn a sketch. And in the sketch, Iโve got a circle, Iโve got the center to the circle, ๐, then weโve got the points ๐ถ and ๐ท. And we can see at the point ๐ถ, we have a tangent. And this tangent is gonna be at right angles to our diameter, as Iโve shown in the diagram.
So now, the first thing you want to do is you want to try and work out what the coordinates of ๐ถ are. And to help us find the coordinates of point ๐ถ, we have a formula. And this formula helps us find the midpoint of a line. And that is, that to find the ๐ฅ-coordinate, we have ๐ฅ one plus ๐ฅ two divided by two. So, itโs the average of the two ๐ฅ coordinates or the mean. And then, the same for the ๐ฆ coordinates. We have ๐ฆ one plus ๐ฆ two divided by two.
Well, to find the coordinates of ๐ถ, weโre not trying to find the midpoint, but in fact we have the midpoint. So, we can use this and our formula to help us find the coordinates of ๐ถ. Because if we substitute in what we know, then the midpoint, negative 11 over two, negative one, must be equal to, then weโve got for the ๐ฅ-coordinate, negative seven plus then the ๐ฅ-coordinate of ๐ถ divided by two, and then for the ๐ฆ-coordinate, seven plus the ๐ฆ-coordinate of ๐ถ divided by two.
So first, weโre gonna start with the ๐ฅ-coordinate and set up an equation. So, we have a negative 11 over two is equal to negative seven plus the ๐ฅ-coordinate of ๐ถ over two. So, first of all, what we can do is multiply each side of the equation by two, and thatโs to remove the fractions. So, we get a negative 11 is equal to negative seven plus the ๐ฅ-coordinate of ๐ถ. So then, what we can do is we can add seven to each side of the equation. And that gives us negative four is equal to the ๐ฅ-coordinate of ๐ถ. So, we now know that the ๐ฅ-coordinate of ๐ถ is going to be negative four.
So, now we can move on and try and find the ๐ฆ-coordinate of ๐ถ. So again, we set up an equation. Weโve got negative one is equal to seven plus the ๐ฆ-coordinate of ๐ถ divided by two. So again, weโre gonna multiply each side of the equation by two to remove the fraction. So, we get negative two is equal to seven plus the ๐ฆ-coordinate of ๐ถ. And then, we subtract seven from each side of the equation. So, we get that negative nine is equal to the ๐ฆ-coordinate of ๐ถ. And thatโs because if we subtract seven from negative two, we get negative nine. So therefore, we found the ๐ฅ and ๐ฆ coordinates of ๐ถ, and they are negative four and negative nine.
Well, why did we do this? Well, we did this because what we want is a point on our tangent line if weโre to find the equation of the tangent to the circle at the point that weโve got, and that point is ๐ถ. Well, letโs have a look at how we work out the equation of a line. Well, one of the general forms of the equation of a straight line that we use is ๐ฆ minus ๐ฆ one is equal to ๐ multiplied by ๐ฅ minus ๐ฅ one, where ๐ฆ one and ๐ฅ one are the coordinates of a point in our line, and ๐ is the slope. Well, we have an ๐ฅ one and ๐ฆ one because we do have a point on the line cause we found out ๐ถ. But we donโt have the slope, so now we need to find out the slope.
Well, to work out the slope, we have a formula and that is that the ๐, our slope, is equal to ๐ฆ two minus ๐ฆ one over ๐ฅ two minus ๐ฅ one. So, what this means is the change in ๐ฆ divided by the change in ๐ฅ. But to do that, weโd need two points on our tangent. But we donโt have two points on our tangent. So, what weโre gonna do now? What weโre gonna do is find out the slope of ๐ถ๐ท, our diameter, and then weโre gonna to use that to find the slope of the tangent.
So, to find the slope of the line ๐ถ๐ท, weโve got our points ๐ถ and ๐ท, and Iโve labelled each of the coordinates ๐ฅ one, ๐ฆ one, ๐ฅ two, ๐ฆ two. So therefore, we have the slope ๐ is equal to seven minus negative nine as the numerator and negative seven minus negative four as the denominator. So therefore, the slope is gonna be equal to negative 16 over three. And we got that because we had seven minus negative nine. Well, if you minus a negative is the same as adding, so seven add nine is 16. And then on the denominator, we had negative seven and then we had again minus negative four, so that negative seven add four is gonna give us negative three. So, it gives us negative 16 over three.
But how is this slope of ๐ถ๐ท gonna help us to find the slope of the tangent? Well, the way itโs gonna help is using this relationship. And thatโs relationship of perpendicular lines, so lines at right angles to each other. We know that if we multiply the slope of two lines that are perpendicular to each other, the result is negative one. So therefore, if we want to find one of these slopes, given the other, then what we do is divide negative one by the other slope.
So, in this case, we could say that ๐ one was gonna be equal to negative one over ๐ two. And the way that we can actually see this is itโs called the negative reciprocal. So, we know that if two lines are perpendicular to each other, their slopes are gonna be the negative reciprocal of each other. So therefore, the slope of our tangent, or ๐ ๐, is gonna be equal to three over 16. Thatโs because we had negative 16 over three. Well, itโs the negative reciprocal, so we changed the sign, so itโs gonna become positive. And the reciprocal means that the numerator and the denominator swap. So, we get three over 16.
So, now that we have the point ๐ถ, negative four, negative nine, and the slope of the tangent to the circle, which is three over 16, we have all the parts we require to put into the general form for the equation of a line. So, we have ๐ฆ minus negative nine is equal to three over 16 ๐ฅ minus negative four. So therefore, what weโre gonna have is ๐ฆ plus nine. Thatโs cause again, weโre subtracting negative turns to a positive. And then, this is equal to three over 16 ๐ฅ. Thatโs cause we multiply three over 16 by ๐ฅ. And then, add 12 over 16. And thatโs because we had ๐ฅ add four cause again weโre here subtracting negative.
So, if you have three over 16 multiplied by four, itโs the same as three over 16 multiplied by four over one. So, three by four is 12. 16 by one is 16. So, we get 12 over 16. So then, what we can do is we can convert positive nine into 36 over four and our 12 over 16 to three over four, so that weโve got them as the same denominator. So, we got ๐ฆ plus 36 over four equals three over 16 ๐ฅ plus three over four. So then, if we subtract 36 over four from each side of the equation to leave ๐ฆ on its own, weโre gonna get ๐ฆ is equal to three over 16 ๐ฅ minus 33 over four. Thatโs cause if we have positive three over four minus 36 over four, we get negative 33 over four.
So therefore, we can say that given that the line ๐ถ๐ท is a diameter of the circle ๐. And the coordinates of the points ๐ and ๐ท are negative 11 over two, negative one and negative seven, seven, respectively. The equation of the tangent to the circle at the point ๐ถ is ๐ฆ equals three over 16 ๐ฅ minus 33 over four.
