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Question
Find the equation of the tangent line to the curve y = x2 − 2x + 7 which is parallel to the line 2x − y + 9 = 0.
Solution
The equation of the given curve is y = x2 − 2x + 7
On differentiating with respect to x, we get:
`"dy"/"dx" = 2x - 2`
The equation of the line is 2x − y + 9 = 0.
2x − y + 9 = 0 ⇒ y = 2x + 9
This is of the form y = mx + c.
∴ Slope of the line = 2
If a tangent is parallel to the line 2x − y + 9 = 0, then the slope of the tangent is equal to the slope of the line.
Therefore, we have:
2 = 2x − 2
⇒ 2x = 4
⇒ x = 2
Now, x = 2
⇒ y = 4 - 4 + 7 = 7
Thus, the equation of the tangent passing through passing through (2, 7) is given by,
⇒ y - 7 = 2 (x - 2)
⇒ y - 2x - 3 = 0
Hence, the equation of the tangent line to the given curve (which is parallel to line 2x - y + 9 = 0) is y - 2x - 3 = 0.
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