Given that 𝐶𝐷 is a diameter of the circle 𝑀, and the coordinates of the points 𝑀 and 𝐷 are negative 11 over two, negative one and negative seven, seven respectively, determine the equation of the tangent to the circle at the point 𝐶.
So, to help us understand what’s going on in this question, I’ve drawn a sketch. And in the sketch, I’ve got a circle, I’ve got the center to the circle, 𝑀, then we’ve got the points 𝐶 and 𝐷. And we can see at the point 𝐶, we have a tangent. And this tangent is gonna be at right angles to our diameter, as I’ve 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 𝐶 are. And to help us find the coordinates of point 𝐶, we have a formula. And this formula helps us find the midpoint of a line. And that is, that to find the 𝑥-coordinate, we have 𝑥 one plus 𝑥 two divided by two. So, it’s the average of the two 𝑥 coordinates or the mean. And then, the same for the 𝑦 coordinates. We have 𝑦 one plus 𝑦 two divided by two.
Well, to find the coordinates of 𝐶, we’re 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 𝐶. Because if we substitute in what we know, then the midpoint, negative 11 over two, negative one, must be equal to, then we’ve got for the 𝑥-coordinate, negative seven plus then the 𝑥-coordinate of 𝐶 divided by two, and then for the 𝑦-coordinate, seven plus the 𝑦-coordinate of 𝐶 divided by two.
So first, we’re gonna start with the 𝑥-coordinate and set up an equation. So, we have a negative 11 over two is equal to negative seven plus the 𝑥-coordinate of 𝐶 over two. So, first of all, what we can do is multiply each side of the equation by two, and that’s to remove the fractions. So, we get a negative 11 is equal to negative seven plus the 𝑥-coordinate of 𝐶. 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 𝑥-coordinate of 𝐶. So, we now know that the 𝑥-coordinate of 𝐶 is going to be negative four.
So, now we can move on and try and find the 𝑦-coordinate of 𝐶. So again, we set up an equation. We’ve got negative one is equal to seven plus the 𝑦-coordinate of 𝐶 divided by two. So again, we’re gonna multiply each side of the equation by two to remove the fraction. So, we get negative two is equal to seven plus the 𝑦-coordinate of 𝐶. And then, we subtract seven from each side of the equation. So, we get that negative nine is equal to the 𝑦-coordinate of 𝐶. And that’s because if we subtract seven from negative two, we get negative nine. So therefore, we found the 𝑥 and 𝑦 coordinates of 𝐶, 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 we’re to find the equation of the tangent to the circle at the point that we’ve got, and that point is 𝐶. Well, let’s 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 𝑦 minus 𝑦 one is equal to 𝑀 multiplied by 𝑥 minus 𝑥 one, where 𝑦 one and 𝑥 one are the coordinates of a point in our line, and 𝑀 is the slope. Well, we have an 𝑥 one and 𝑦 one because we do have a point on the line cause we found out 𝐶. But we don’t 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 𝑀, our slope, is equal to 𝑦 two minus 𝑦 one over 𝑥 two minus 𝑥 one. So, what this means is the change in 𝑦 divided by the change in 𝑥. But to do that, we’d need two points on our tangent. But we don’t have two points on our tangent. So, what we’re gonna do now? What we’re gonna do is find out the slope of 𝐶𝐷, our diameter, and then we’re gonna to use that to find the slope of the tangent.
So, to find the slope of the line 𝐶𝐷, we’ve got our points 𝐶 and 𝐷, and I’ve labelled each of the coordinates 𝑥 one, 𝑦 one, 𝑥 two, 𝑦 two. So therefore, we have the slope 𝑀 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 𝐶𝐷 gonna help us to find the slope of the tangent? Well, the way it’s gonna help is using this relationship. And that’s 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 𝑚 one was gonna be equal to negative one over 𝑚 two. And the way that we can actually see this is it’s 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 𝑚 𝑇, is gonna be equal to three over 16. That’s because we had negative 16 over three. Well, it’s the negative reciprocal, so we changed the sign, so it’s 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 𝐶, 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 𝑦 minus negative nine is equal to three over 16 𝑥 minus negative four. So therefore, what we’re gonna have is 𝑦 plus nine. That’s cause again, we’re subtracting negative turns to a positive. And then, this is equal to three over 16 𝑥. That’s cause we multiply three over 16 by 𝑥. And then, add 12 over 16. And that’s because we had 𝑥 add four cause again we’re here subtracting negative.
So, if you have three over 16 multiplied by four, it’s 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 we’ve got them as the same denominator. So, we got 𝑦 plus 36 over four equals three over 16 𝑥 plus three over four. So then, if we subtract 36 over four from each side of the equation to leave 𝑦 on its own, we’re gonna get 𝑦 is equal to three over 16 𝑥 minus 33 over four. That’s 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 𝐶𝐷 is a diameter of the circle 𝑀. And the coordinates of the points 𝑀 and 𝐷 are negative 11 over two, negative one and negative seven, seven, respectively. The equation of the tangent to the circle at the point 𝐶 is 𝑦 equals three over 16 𝑥 minus 33 over four.