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.

Answer 1

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.