Advertisements
Advertisements
Question
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.
Solution
Le P(x1, y1) be the point on the curve y = x2 + 5 where the tangent is parallel to the line 4x – y + 1 = 0.
Differentiating w.r.t.x we get
`dy/dx = d/dx(x^2 + 5)` = 2x + 0 = 2x
∴ `(dy/dx)_(at(x_1.y_1)` = 2x1
Slope of the tangent at (x1, y1)
Let m1 = 2x1,
The slope of line 4x – y + 1 = 0 is
m2 = `(-4)/(-1)` = 4
∵ The tangent at P(x1, y1) is parallel to the line
4x – y + 1 = 0, m1 = m2
∴ 2x1 = 4
∴ x1 = 2
Since (x1, y1) lies on the curve y = x2 + 5, y1 = x12 + 5
∴ y1 = (2)2 + 5 = 9 ...[∵ x1 = 2]
∴ The coordinates of the point are (2, 9) and the slope of the tangent= m1 = m2 = 4
∴ Equation of the tangent is y – 9 = 4(x – 2)
∴ y – 9 = 4(x – 2)
∴ y – 9 = 4x – 8
∴ 4x – y + 1 = 0
Slope of the normal = `(-1)/m_1 = -1/4`
∴ Equation of normal at (2, 9) is y – 9 = `(-1)/4(x - 2)`
∴ 4y – 36 = –x + 2
∴ x + 4y – 38 = 0
Hence, the equation of tangent and normal are 4x – y + 1 = 0 and x + 4y – 38 = 0 respectively.
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.
`1/x^2`
Find the derivative of the following function from first principle.
`(x+1)/(x -1)`
Find the derivative of the following function from first principle:
sin (x + 1)
Find the derivative of the following function from first principle:
`cos (x - pi/8)`
Find the equation of tangent and normal to the curve at the given points on it.
2x2 + 3y2 = 5 at (1, 1)
Find the equation of tangent and normal to the curve at the given points on it.
x2 + y2 + xy = 3 at (1, 1)
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.
Find the equations of tangent and normal to the curve y = 3x2 - 3x - 5 where the tangent is parallel to the line 3x − y + 1 = 0.
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 0 < η < 1, then demand is
Choose the correct alternative.
If f(x) = 3x3 - 9x2 - 27x + 15 then
Fill in the blank:
The slope of tangent at any point (a, b) is called as _______.
Fill in the blank:
If f(x) = x - 3x2 + 3x - 100, x ∈ R 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
Find the equation of tangent and normal to the following curve.
xy = c2 at `("ct", "c"/"t")` where t is parameter.
Find the equation of tangent and normal to the following curve.
y = x2 + 4x at the point whose ordinate is -3.
Find the equation of tangent and normal to the following curve.
y = x3 - x2 - 1 at the point whose abscissa is -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 ______
Slope of the tangent to the curve y = 6 – x2 at (2, 2) is ______.
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
