Math CentralQuandaries & Queries


Question from Reuchen, a student:

Find equations of the lines tangent to y^2=4x and containing (-2,1).

Hi Reuchen,

Suppose that $P$ is a point on the curve where the tangent line to the curve at $P$ passes through $(-2, 1).$ Let $P$ have coordinates $(a, b)$ then since $P$ is on the curve we know that $b^2 = 4a.$

  1. Find the slope of the line containing $P = (a, b)$ and $(-2, 1).$

  2. Differentiate both sides of $y^2 = 4 x$ with respect to $x$ and solve for $y'.$ Evaluate $y'$ at $P.$

The value of $y'$ at $P$ and the slope you found in part 1. are both the slope of the tangent line to the curve at $P.$ Substitute for $a$ using $b^2 = 4a$ and solve for b.


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