Advertisements
Advertisements
Question
Find the equations of tangent and normal to the curve y = x2 + 5 where the tangent is parallel to the line 4x − y + 1 = 0.
Solution
Equation of the curve is y = x2 + 5
Differentiating w.r.t. x, we get
`"dy"/"dx"` = 2x
Slope of the tangent at P(x1, y1) is
`("dy"/"dx")_(("x"_1,"x"_2)` = 2x1
According to the given condition, the tangent is parallel to 4x – y + 1 = 0
Now, slope of the line 4x – y + 1 = 0 is 4.
∴ Slope of the tangent = `"dy"/"dx" = 4`
∴ 2x1 = 4
∴ x1 = 2
P(x1, y1) lies on the curve y = x2 + 5
∴ y1 = (2)2 + 5
∴ y1 = 9
∴ The point on the curve is (2, 9).
∴ Equation of the tangent at (2, 9) is
∴ (y – 9) = 4(x – 2)
∴ y – 9 = 4x – 8
∴ 4x - y + 1 = 0
Slope of the normal at (2, 9) is `1/("dy"/"dx")_((2,9)` = `(- 1)/4`
∴ Equation of the normal of (2, 9) is
(y - 9) = `(-1)/4`(x - 2)
∴ 4y - 36 = - x + 2
∴ x + 4y - 38 = 0
APPEARS IN
RELATED QUESTIONS
Find the derivative of the following function from first principle.
x3 – 27
Find the derivative of the following function from first principle:
−x
Find the equation of tangent and normal to the curve at the given points on it.
x2 + y2 + xy = 3 at (1, 1)
Choose the correct alternative.
The equation of tangent to the curve x2 + y2 = 5 where the tangent is parallel to the line 2x – y + 1 = 0 are
Choose the correct alternative.
If f(x) = 3x3 - 9x2 - 27x + 15 then
Fill in the blank:
If f(x) = x - 3x2 + 3x - 100, x ∈ R then f''(x) is ______
Fill in the blank:
If f(x) = `7/"x" - 3`, x ∈ R x ≠ 0 then f ''(x) is ______
State whether the following statement is True or False:
The equation of tangent to the curve y = 4xex at `(-1, (- 4)/"e")` is ye + 4 = 0
State whether the following statement is True or False:
x + 10y + 21 = 0 is the equation of normal to the curve y = 3x2 + 4x - 5 at (1, 2).
Find the equation of normal to the curve y = `sqrt(x - 3)` which is perpendicular to the line 6x + 3y – 4 = 0.
The slope of the tangent to the curve y = x3 – x2 – 1 at the point whose abscissa is – 2, is ______.
The slope of the tangent to the curve x = `1/"t"`, y = `"t" - 1/"t"`, at t = 2 is ______
Find the equation of tangent to the curve x2 + y2 = 5, where the tangent is parallel to the line 2x – y + 1 = 0
Find the equation of tangent to the curve y = x2 + 4x at the point whose ordinate is – 3
Find the equation of tangent and normal to the curve y = x2 + 5 where the tangent is parallel to the line 4x – y + 1 = 0.
y = ae2x + be-3x is a solution of D.E. `(d^2y)/dx^2 + dy/dx + by = 0`
Find the equations of tangent and normal to the curve y = 6 - x2 where the normal is parallel to the line x - 4y + 3 = 0
