Find the equation of the tangent through a line y = 5x⁴ + 9x⁶ =0 at ( 1,2). Hint : Use derivatives
Answers
Answer:
The equation of the given curve is y=x
2
−2x+7.
On differentiating with respect to x, we get:
dx
dy
=2x−2
The equation of the line is 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, at x=2
⇒y=2
2
−2×2+7=7
Thus, the equation of the tangent 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.
Step-by-step explanation:
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Step-by-step explanation:
Differentiate the equation w.r.t. x, we get.
dy/dx = 20x³ +54x5.
Put the value of (x, y) = (1, 2) in the equation, we get.
dy/dx = 20(1)³ + 54(1)5.
dy/dx=
= 20 + 54.
dy/dx = 74.
As we know that, dy/dx is the slope of the equation = m.
→ m = 74.
As we know that,
Formula of equation of tangent.
(y - y₁) = m(x - x₁).
Put the values in the equation, we get.
(y-2) = 74(x - 1).
⇒ y - 2 = 74x - 74.
→ 74x - 74-y + 2 = 0.
Equation of tangent.
(1) = Equation of tangent to the curve y = f(x) at p(x₁, y₁) is (y - y₁) = m(x - x₁).
(2) The tangent at (x₁, y₁) is parallel to x-axis : dy/dx = 0.
(3) = The tangent at (x₁, y₁) is parallel to
y-axis : dy/dx = 0.
(4) = The tangent line makes equal angles with the axis : dy/dx = + 1.