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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 ?
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Solution
The equation of given curve is `y=x^2-2x+7`
On differentiating with respect to `x,` we get:
`(dy)/(dx)=2x-2`
The equation of the line is 2x - y + 9 = 0
`rArry=2x+9`
This is of the form `y=mx+c`
`therefore "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=x^2-2x+7`
`rArry=4-4+7`
Thus, the equation of the tangent passing through (2, 7) is given by,
`y-y_1=m(x-x_1)`
y - 7 = 2(x - 2)
`rArry-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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