Science:Math Exam Resources/Courses/MATH100/December 2018/Question 04/Solution 1

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Let be a point on the curve. Then

To find the slope of the tangent line passing through , use implicit differentiation. Differentiate both sides of the equation of the curve with respect to . By the chain rule,

Let be the slope of the tangent line at point . Assume . Then

Using the point-slope formula, the tangent line equation to the curve at the point can be written as

Let this tangent line pass through . Putting and ,

Together with Equation (1) we can solve for the point. or

Now consider if , then The slope of the tangent line DNE. It follows that the tangent line is a vertical line, either or . Neither of them passes through the point

Therefore, the points on the curve that satisfy the requirements are .