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    Hey guys, I've been recapping Core Maths and I stumbled upon a C1 Q. It asks to find the coordinates of the point at which the line y = 9x-9 is a tangent to the curve but I received two separate coordinates, one of which I got the answer for and another is another point. My question is that how would you know which point they were asking for, especially without knowing the multiple choice answers?

    http://i-want-to-study-engineering.o...ths_c1_q10iii/
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    If you have found the point on the curve where the gradient is 9. There can be more than one point. You have to consider the geometry and see which answer for the value of x is correct.
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    The equation of the line is in the form y=mx+c so what is the gradient? Make this equal to dy/dx and solve the quadratic. To check which one of the two values it is; plug them both through the tangent line equation and the cubic. Which ever x value gives the same answer to both, that's the x and y co-ordinate. This is because they are tangent therefore only 1 point of intersection
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    The equation of the line is in the form y=mx+c so what is the gradient? Make this equal to dy/dx and solve the quadratic. To check which one of the two values it is; plug them both through the tangent line equation and the cubic. Which ever x value gives the same answer to both, that's the x and y co-ordinate. This is because they are tangent therefore only 1 point of intersection.

    I kinda see what you mean. I got the coordinates (-2,-27) and (0,-5) and then substituting each value of x, the y value was the same when x = -2 but then I kinda don't get what you mean for your reason. Isn't the coordinates two points on the curve where the line y = 9x-9 a tangent? I'm a bit confused .
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    (Original post by Logic938)

    I kinda see what you mean. I got the coordinates (-2,-27) and (0,-5) and then substituting each value of x, the y value was the same when x = -2 but then I kinda don't get what you mean for your reason. Isn't the coordinates two points on the curve where the line y = 9x-9 a tangent? I'm a bit confused .
    The line can only have one point of intersection with the curve as it is tangent. When you find the two values of x from the quadratic, it is saying that there are 2 points on the cubic where the gradient is equal to 9. So while y=9x-9 is a tangent at one point, the line y=9x-5 is tangent to the other (notice the same gradient, different point of intersection). By running the two x-values through y=9x-9 and the cubic, you are looking which of the two satisfies the linear equation you are given. This means x=0 gives the co-ordinates of the tangent for y=9x-5.
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    (Original post by RDKGames)
    The line can only have one point of intersection with the curve as it is tangent. When you find the two values of x from the quadratic, it is saying that there are 2 points on the cubic where the gradient is equal to 9. So while y=9x-9 is a tangent at one point, the line y=9x-5 is tangent to the other (notice the same gradient, different point of intersection). By running the two x-values through y=9x-9 and the cubic, you are looking which of the two satisfies the linear equation you are given. This means x=0 gives the co-ordinates of the tangent for y=9x-5.
    That's definitely clarified my confusion and I checked a sketch of the two equations and can visually see why.

    Thanks again!
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    (Original post by RDKGames)
    The line can only have one point of intersection with the curve as it is tangent.
    Well. I know what you mean but it's not really true as I'm sure you know. A tangent could intersect a curve again at some other point.
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    (Original post by B_9710)
    Well. I know what you mean but it's not really true as I'm sure you know. A tangent could intersect a curve again at some other point.
    Ah yes, you are right. I was thinking of a tangent on a quadratic for some reason lol. x_x
 
 
 
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