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Question
Find a point on the curve y = (x − 2)2 at which the tangent is parallel to the chord joining the points (2, 0) and (4, 4).
Solution
If a tangent is parallel to the chord joining the points (2, 0) and (4, 4), then the slope of the tangent = the slope of the chord.
The slope of the chord is `(4 - 0)/(4 - 2) = 4/2 = 2`.
Now, the slope of the tangents to the given curve at a point (x, y) is given by,
`dy/dx = 2(x - 2)`
Since the slope of the tangent = slope of the chord, we have:
2(x – 2) = 2
`\implies` x – 2 = 1
`\implies` x = 3
When x = 3, y = (3 – 2)2 = 1.
Hence, the required point is (3, 1).
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