Advertisements
Advertisements
Question
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.
Solution
Equation of the curve is y = 3x2 - 3x - 5
Differentiating w.r.t. x, we get
`"dy"/"dx"` = 6x - 3
Slope of the tangent at P(x1, y1) is
`("dy"/"dx")_(("x"_1,"x"_2)` = 6x1 - 3
According to the given condition, the tangent is parallel to 3x – y + 1 = 0
Now, slope of the line 3x – y + 1 = 0 is 3.
∴ Slope of the tangent = `"dy"/"dx" = 3`
∴ 6x1 - 3 = 3
∴ x1 = 1
P(x1, y1) lies on the curve y = 3x2 - 3x - 5
∴ y1 = 3(1)2 - 3(1) - 5
∴ y1 = - 5
∴ The point on the curve is (1, -5).
∴ Equation of the tangent at (1, -5) is
∴ (y + 5) = 3(x – 1)
∴ y + 5 = 3x – 3
∴ 3x - y - 8 = 0
Slope of the normal at (1, -5) is `(-1)/("dy"/"dx")_((1,-5)` = `(- 1)/3`
∴ Equation of the normal of (1, -5) is
(y + 5) = `(-1)/3`(x - 1)
∴ 3y + 15 = - x + 1
∴ x + 3y + 14 = 0
APPEARS IN
RELATED QUESTIONS
Find the derivative of the following function from first principle:
(–x)–1
Find the derivative of the following function from first principle:
sin (x + 1)
Find the equation of tangent and normal to the curve at the given points on it.
y = 3x2 - x + 1 at (1, 3)
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.
Fill in the blank:
The slope of tangent at any point (a, b) is called as _______.
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.
x = `1/"t", "y" = "t" - 1/"t"`, at t = 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 equations of tangent and normal to the curve y = 3x2 – x + 1 at the point (1, 3) on it
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
Slope of the tangent to the curve y = 6 – x2 at (2, 2) is ______.
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.
