Chapters
Chapter 2: Functions
Chapter 3: Binary Operations
Chapter 4: Inverse Trigonometric Functions
Chapter 5: Algebra of Matrices
Chapter 6: Determinants
Chapter 7: Adjoint and Inverse of a Matrix
Chapter 8: Solution of Simultaneous Linear Equations
Chapter 9: Continuity
Chapter 10: Differentiability
Chapter 11: Differentiation
Chapter 12: Higher Order Derivatives
Chapter 13: Derivative as a Rate Measurer
Chapter 14: Differentials, Errors and Approximations
Chapter 15: Mean Value Theorems
Chapter 16: Tangents and Normals
Chapter 17: Increasing and Decreasing Functions
Chapter 18: Maxima and Minima
Chapter 19: Indefinite Integrals
Chapter 20: Definite Integrals
Chapter 21: Areas of Bounded Regions
Chapter 22: Differential Equations
Chapter 23: Algebra of Vectors
Chapter 24: Scalar Or Dot Product
Chapter 25: Vector or Cross Product
Chapter 26: Scalar Triple Product
Chapter 27: Direction Cosines and Direction Ratios
Chapter 28: Straight Line in Space
Chapter 29: The Plane
Chapter 30: Linear programming
Chapter 31: Probability
Chapter 32: Mean and Variance of a Random Variable
Chapter 33: Binomial Distribution

Chapter 16: Tangents and Normals
RD Sharma solutions for Class 12 Maths Chapter 16 Tangents and Normals Exercise 16.1 [Pages 10 - 11]
Find the slope of the tangent and the normal to the following curve at the indicted point \[y = \sqrt{x^3} \text { at } x = 4\] ?
Find the slope of the tangent and the normal to the following curve at the indicted point \[y = \sqrt{x} \text { at }x = 9\] ?
Find the slope of the tangent and the normal to the following curve at the indicted point y = x3 − x at x = 2 ?
Find the slope of the tangent and the normal to the following curve at the indicted point y = 2x2 + 3 sin x at x = 0 ?
Find the slope of the tangent and the normal to the following curve at the indicted point x = a (θ − sin θ), y = a(1 − cos θ) at θ = −π/2 ?
Find the slope of the tangent and the normal to the following curve at the indicted point x = a cos3 θ, y = a sin3 θ at θ = π/4 ?
Find the slope of the tangent and the normal to the following curve at the indicted point x = a (θ − sin θ), y = a(1 − cos θ) at θ = π/2 ?
Find the slope of the tangent and the normal to the following curve at the indicted point y = (sin 2x + cot x + 2)2 at x = π/2 ?
Find the slope of the tangent and the normal to the following curve at the indicted point x2 + 3y + y2 = 5 at (1, 1) ?
Find the slope of the tangent and the normal to the following curve at the indicted point xy = 6 at (1, 6) ?
Find the values of a and b if the slope of the tangent to the curve xy + ax + by = 2 at (1, 1) is 2 ?
If the tangent to the curve y = x3 + ax + b at (1, − 6) is parallel to the line x − y + 5 = 0, find a and b ?
Find a point on the curve y = x3 − 3x where the tangent is parallel to the chord joining (1, −2) and (2, 2) ?
Find the points on the curve y = x3 − 2x2 − 2x at which the tangent lines are parallel to the line y = 2x− 3 ?
Find the points on the curve y2 = 2x3 at which the slope of the tangent is 3 ?
Find the points on the curve xy + 4 = 0 at which the tangents are inclined at an angle of 45° with the x-axis ?
Find the point on the curve y = x2 where the slope of the tangent is equal to the x-coordinate of the point ?
At what points on the circle x2 + y2 − 2x − 4y + 1 = 0, the tangent is parallel to x-axis?
At what point of the curve y = x2 does the tangent make an angle of 45° with the x-axis?
Find the points on the curve y = 3x2 − 9x + 8 at which the tangents are equally inclined with the axes ?
At what points on the curve y = 2x2 − x + 1 is the tangent parallel to the line y = 3x + 4?
Find the point on the curve y = 3x2 + 4 at which the tangent is perpendicular to the line whose slop is \[- \frac{1}{6}\] ?
Find the points on the curve x2 + y2 = 13, the tangent at each one of which is parallel to the line 2x + 3y = 7 ?
Find the points on the curve 2a2y = x3 − 3ax2 where the tangent is parallel to x-axis ?
At what points on the curve y = x2 − 4x + 5 is the tangent perpendicular to the line 2y + x = 7?
Find the points on the curve \[\frac{x^2}{4} + \frac{y^2}{25} = 1\] at which the tangent is parallel to the x-axis ?
Find the points on the curve\[\frac{x^2}{4} + \frac{y^2}{25} = 1\] at which the tangent is parallel to the y-axis ?
Find the points on the curve x2 + y2 − 2x − 3 = 0 at which the tangents are parallel to the x-axis ?
Find the points on the curve \[\frac{x^2}{9} + \frac{y^2}{16} = 1\] at which the tangent is parallel to x-axis ?
Find the points on the curve \[\frac{x^2}{9} + \frac{y^2}{16} = 1\] at which the tangent is parallel to y-axis ?
Who that the tangents to the curve y = 7x3 + 11 at the points x = 2 and x = −2 are parallel ?
Find the points on the curve y = x3 where the slope of the tangent is equal to the x-coordinate of the point ?
RD Sharma solutions for Class 12 Maths Chapter 16 Tangents and Normals Exercise 16.2 [Pages 27 - 29]
Find the equation of the tangent to the curve \[\sqrt{x} + \sqrt{y} = a\] at the point \[\left( \frac{a^2}{4}, \frac{a^2}{4} \right)\] ?
Find the equation of the normal to y = 2x3 − x2 + 3 at (1, 4) ?
Find the equation of the tangent and the normal to the following curve at the indicated point y = x4 − bx3 + 13x2 − 10x + 5 at (0, 5) ?
Find the equation of the tangent and the normal to the following curve at the indicated point y = x4 − 6x3 + 13x2 − 10x + 5 at x = 1?
Find the equation of the tangent and the normal to the following curve at the indicated point y = x2 at (0, 0) ?
Find the equation of the tangent and the normal to the following curve at the indicated point y = 2x2 − 3x − 1 at (1, −2) ?
Find the equation of the tangent and the normal to the following curve at the indicated point \[y^2 = \frac{x^3}{4 - x}at \left( 2, - 2 \right)\] ?
Find the equation of the tangent and the normal to the following curve at the indicated point y = x2 + 4x + 1 at x = 3 ?
Find the equation of the tangent and the normal to the following curve at the indicated point\[\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \text{ at }\left( a\cos\theta, b\sin\theta \right)\] ?
Find the equation of the tangent and the normal to the following curve at the indicated point \[\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \text { at } \left( a\sec\theta, b\tan\theta \right)\] ?
Find the equation of the tangent and the normal to the following curve at the indicated point y2 = 4ax at \[\left( \frac{a}{m^2}, \frac{2a}{m} \right)\] ?
Find the equation of the tangent and the normal to the following curve at the indicated point \[c^2 \left( x^2 + y^2 \right) = x^2 y^2 \text { at }\left( \frac{c}{\cos\theta}, \frac{c}{\sin\theta} \right)\] ?
Find the equation of the tangent and the normal to the following curve at the indicated point xy = c2 at \[\left( ct, \frac{c}{t} \right)\] ?
Find the equation of the tangent and the normal to the following curve at the indicated point \[\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \text { at } \left( x_1 , y_1 \right)\] ?
Find the equation of the tangent and the normal to the following curve at the indicated point \[\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \text { at } \left( x_0 , y_0 \right)\] ?
Find the equation of the tangent and the normal to the following curve at the indicated point \[x^\frac{2}{3} + y^\frac{2}{3}\] = 2 at (1, 1) ?
Find the equation of the tangent and the normal to the following curve at the indicated point x2 = 4y at (2, 1) ?
Find the equation of the tangent and the normal to the following curve at the indicated point y2 = 4x at (1, 2) ?
Find the equation of the tangent and the normal to the following curve at the indicated point 4x2 + 9y2 = 36 at (3cosθ, 2sinθ) ?
Find the equation of the tangent and the normal to the following curve at the indicated point y2 = 4ax at (x1, y1)?
Find the equation of the tangent and the normal to the following curve at the indicated point \[\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \text { at } \left( \sqrt{2}a, b \right)\] ?
Find the equation of the tangent to the curve x = θ + sin θ, y = 1 + cos θ at θ = π/4 ?
Find the equation of the tangent and the normal to the following curve at the indicated points x = θ + sinθ, y = 1 + cosθ at θ = \[\frac{\pi}{2}\] ?
Find the equation of the tangent and the normal to the following curve at the indicated points \[x = \frac{2 a t^2}{1 + t^2}, y = \frac{2 a t^3}{1 + t^2}\text { at } t = \frac{1}{2}\] ?
Find the equation of the tangent and the normal to the following curve at the indicated points x = at2, y = 2at at t = 1 ?
Find the equation of the tangent and the normal to the following curve at the indicated points x = asect, y = btant at t ?
Find the equation of the tangent and the normal to the following curve at the indicated points x = a(θ + sinθ), y = a(1 − cosθ) at θ ?
Find the equation of the tangent and the normal to the following curve at the indicated points x = 3cosθ − cos3θ, y = 3sinθ − sin3θ ?
Find the equation of the normal to the curve x2 + 2y2 − 4x − 6y + 8 = 0 at the point whose abscissa is 2 ?
Find the equation of the normal to the curve ay2 = x3 at the point (am2, am3) ?
The equation of the tangent at (2, 3) on the curve y2 = ax3 + b is y = 4x − 5. Find the values of a and b ?
Find the equation of the tangent line to the curve y = x2 + 4x − 16 which is parallel to the line 3x − y + 1 = 0 ?
Find an equation of normal line to the curve y = x3 + 2x + 6 which is parallel to the line x+ 14y + 4 = 0 ?
Determine the equation(s) of tangent (s) line to the curve y = 4x3 − 3x + 5 which are perpendicular to the line 9y + x + 3 = 0 ?
Find the equation of a normal to the curve y = x loge x which is parallel to the line 2x − 2y + 3 = 0 ?
Find the equation of the tangent line to the curve y = x2 − 2x + 7 which is parallel to the line 2x − y + 9 = 0 ?
Find the equation of the tangent line to the curve y = x2 − 2x + 7 which perpendicular to the line 5y − 15x = 13. ?
Find the equations of all lines having slope 2 and that are tangent to the curve \[y = \frac{1}{x - 3}, x \neq 3\] ?
Find the equations of all lines of slope zero and that are tangent to the curve \[y = \frac{1}{x^2 - 2x + 3}\] ?
Find the equation of the tangent to the curve \[y = \sqrt{3x - 2}\] which is parallel to the 4x − 2y + 5 = 0 ?
Find the equation of the tangent to the curve x2 + 3y − 3 = 0, which is parallel to the line y= 4x − 5 ?
Prove that \[\left( \frac{x}{a} \right)^n + \left( \frac{y}{b} \right)^n = 2\] touches the straight line \[\frac{x}{a} + \frac{y}{b} = 2\] for all n ∈ N, at the point (a, b) ?
Find the equation of the tangent to the curve x = sin 3t, y = cos 2t at
\[t = \frac{\pi}{4}\] ?
At what points will be tangents to the curve y = 2x3 − 15x2 + 36x − 21 be parallel to x-axis ? Also, find the equations of the tangents to the curve at these points ?
Find the equation of the tangents to the curve 3x2 – y2 = 8, which passes through the point (4/3, 0) ?
RD Sharma solutions for Class 12 Maths Chapter 16 Tangents and Normals Exercise 16.3 [Pages 40 - 41]
Find the angle of intersection of the following curve y2 = x and x2 = y ?
Find the angle of intersection of the following curve y = x2 and x2 + y2 = 20 ?
Find the angle of intersection of the following curve 2y2 = x3 and y2 = 32x ?
Find the angle of intersection of the following curve x2 + y2 − 4x − 1 = 0 and x2 + y2 − 2y − 9 = 0 ?
Find the angle of intersection of the following curve \[\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\] and x2 + y2 = ab ?
Find the angle of intersection of the following curve x2 + 4y2 = 8 and x2 − 2y2 = 2 ?
Find the angle of intersection of the following curve x2 = 27y and y2 = 8x ?
Find the angle of intersection of the following curve x2 + y2 = 2x and y2 = x ?
Find the angle of intersection of the following curve y = 4 − x2 and y = x2 ?
Show that the following set of curve intersect orthogonally y = x3 and 6y = 7 − x2 ?
Show that the following set of curve intersect orthogonally x3 − 3xy2 = −2 and 3x2y − y3 = 2 ?
Show that the following set of curve intersect orthogonally x2 + 4y2 = 8 and x2 − 2y2 = 4 ?
Show that the following curve intersect orthogonally at the indicated point x2 = 4y and 4y + x2 = 8 at (2, 1) ?
Show that the following curve intersect orthogonally at the indicated point x2 = y and x3 + 6y = 7 at (1, 1) ?
Show that the following curve intersect orthogonally at the indicated point y2 = 8x and 2x2 + y2 = 10 at \[\left( 1, 2\sqrt{2} \right)\] ?
Show that the curves 4x = y2 and 4xy = k cut at right angles, if k2 = 512 ?
Show that the curves 2x = y2 and 2xy = k cut at right angles, if k2 = 8 ?
Prove that the curves xy = 4 and x2 + y2 = 8 touch each other ?
Prove that the curves y2 = 4x and x2 + y2 - 6x + 1 = 0 touch each other at the point (1, 2) ?
Find the condition for the following set of curve to intersect orthogonally \[\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \text { and } xy = c^2\] ?
Find the condition for the following set of curve to intersect orthogonally \[\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \text { and } \frac{x^2}{A^2} - \frac{y^2}{B^2} = 1\] ?
Show that the curves \[\frac{x^2}{a^2 + \lambda_1} + \frac{y^2}{b^2 + \lambda_1} = 1 \text { and } \frac{x^2}{a^2 + \lambda_2} + \frac{y^2}{b^2 + \lambda_2} = 1\] intersect at right angles ?
If the straight line xcos \[\alpha\] +y sin \[\alpha\] = p touches the curve \[\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\] then prove that a2cos2 \[\alpha\] \[-\] b2sin2 \[\alpha\] = p2 ?
RD Sharma solutions for Class 12 Maths Chapter 16 Tangents and Normals [Pages 41 - 42]
Find the point on the curve y = x2 − 2x + 3, where the tangent is parallel to x-axis ?
Find the slope of the tangent to the curve x = t2 + 3t − 8, y = 2t2 − 2t − 5 at t = 2 ?
If the tangent line at a point (x, y) on the curve y = f(x) is parallel to x-axis, then write the value of \[\frac{dy}{dx}\] ?
Write the value of \[\frac{dy}{dx}\] , if the normal to the curve y = f(x) at (x, y) is parallel to y-axis ?
If the tangent to a curve at a point (x, y) is equally inclined to the coordinates axes then write the value of \[\frac{dy}{dx}\] ?
If the tangent line at a point (x, y) on the curve y = f(x) is parallel to y-axis, find the value of \[\frac{dx}{dy}\] ?
Find the slope of the normal at the point 't' on the curve \[x = \frac{1}{t}, y = t\] ?
Write the coordinates of the point on the curve y2 = x where the tangent line makes an angle \[\frac{\pi}{4}\] with x-axis ?
Write the angle made by the tangent to the curve x = et cos t, y = et sin t at \[t = \frac{\pi}{4}\] with the x-axis ?
Write the equation of the normal to the curve y = x + sin x cos x at \[x = \frac{\pi}{2}\] ?
Find the coordinates of the point on the curve y2 = 3 − 4x where tangent is parallel to the line 2x + y− 2 = 0 ?
Write the equation on the tangent to the curve y = x2 − x + 2 at the point where it crosses the y-axis ?
Write the angle between the curves y2 = 4x and x2 = 2y − 3 at the point (1, 2) ?
Write the angle between the curves y = e−x and y = ex at their point of intersections ?
Write the slope of the normal to the curve \[y = \frac{1}{x}\] at the point \[\left( 3, \frac{1}{3} \right)\] ?
Write the coordinates of the point at which the tangent to the curve y = 2x2 − x + 1 is parallel to the line y = 3x + 9 ?
Write the equation of the normal to the curve y = cos x at (0, 1) ?
Write the equation of the tangent drawn to the curve \[y = \sin x\] at the point (0,0) ?
RD Sharma solutions for Class 12 Maths Chapter 16 Tangents and Normals [Pages 42 - 44]
The equation to the normal to the curve y = sin x at (0, 0) is ___________ .
x = 0
y = 0
x + y = 0
x − y = 0
The equation of the normal to the curve y = x + sin x cos x at x = `π/2` is ___________ .
x = 2
x = π
x + π = 0
2x = π
The equation of the normal to the curve y = x(2 − x) at the point (2, 0) is ________________ .
x − 2y = 2
x − 2y + 2 = 0
2x + y = 4
2x + y − 4 = 0
The point on the curve y2 = x where tangent makes 45° angle with x-axis is ______________ .
(1/2, 1/4)
(1/4, 1/2)
(4, 2)
(1, 1)
If the tangent to the curve x = a t2, y = 2 at is perpendicular to x-axis, then its point of contact is _____________ .
(a, a)
(0, a)
(0, 0)
(a, 0)
The point on the curve y = x2 − 3x + 2 where tangent is perpendicular to y = x is ________________ .
(0, 2)
(1, 0)
(−1, 6)
(2, −2)
The point on the curve y2 = x where tangent makes 45° angle with x-axis is ____________________ .
(1/2, 1/4)
(1/4, 1/2)
(4, 2)
(1, 1)
The point at the curve y = 12x − x2 where the slope of the tangent is zero will be _____________ .
(0, 0)
(2, 16)
(3, 9)
none of these
The angle between the curves y2 = x and x2 = y at (1, 1) is ______________ .
\[\tan^{- 1} \frac{4}{3}\]
\[\tan^{- 1} \frac{3}{4}\]
90°
45°
The equation of the normal to the curve 3x2 − y2 = 8 which is parallel to x + 3y = 8 is ____________ .
x + 3y = 8
x + 3y + 8 = 0
x + 3y ± 8 = 0
x + 3y = 0
The equations of tangent at those points where the curve y = x2 − 3x + 2 meets x-axis are _______________ .
x − y + 2 = 0 = x − y − 1
x + y − 1 = 0 = x − y − 2
x − y − 1 = 0 = x − y
x − y = 0 = x + y
The slope of the tangent to the curve x = t2 + 3 t − 8, y = 2t2 − 2t − 5 at point (2, −1) is ________________ .
22/7
6/7
`-6`
none of these
At what point the slope of the tangent to the curve x2 + y2 − 2x − 3 = 0 is zero
(3, 0), (−1, 0)
(3, 0), (1, 2)
(−1, 0), (1, 2)
(1, 2), (1, −2)
The angle of intersection of the curves xy = a2 and x2 − y2 = 2a2 is ______________ .
0°
45°
90°
none of these
If the curve ay + x2 = 7 and x3 = y cut orthogonally at (1, 1), then a is equal to _____________ .
1
`-6`
6
0
If the line y = x touches the curve y = x2 + bx + c at a point (1, 1) then _____________ .
b = 1, c = 2
b = −1, c = 1
b = 2, c = 1
b = −2, c = 1
The slope of the tangent to the curve x = 3t2 + 1, y = t3 −1 at x = 1 is ___________ .
1/2
0
`-2`
∞
The curves y = aex and y = be−x cut orthogonally, if ___________ .
a = b
a = −b
ab = 1
ab = 2
The equation of the normal to the curve x = a cos3 θ, y = a sin3 θ at the point θ = π/4 is __________ .
x = 0
y = 0
x = y
x + y = a
If the curves y = 2 ex and y = ae−x intersect orthogonally, then a = _____________ .
1/2
−1/2
2
2e2
The point on the curve y = 6x − x2 at which the tangent to the curve is inclined at π/4 to the line x + y= 0 is __________ .
(−3, −27)
(3, 9)
(7/2, 35/4)
(0, 0)
The angle of intersection of the parabolas y2 = 4 ax and x2 = 4ay at the origin is ____________ .
π/6
π/3
π/2
π/4
The angle of intersection of the curves y = 2 sin2 x and y = cos 2 x at \[x = \frac{\pi}{6}\] is ____________ .
π/4
π/2
π/3
none of these
Any tangent to the curve y = 2x7 + 3x + 5 __________________ .
is parallel to x-axis
is parallel to y-axis
makes an acute angle with x-axis
makes an obtuse angle with x-axis
The point on the curve 9y2 = x3, where the normal to the curve makes equal intercepts with the axes is
(a) \[\left( 4, \frac{8}{3} \right)\]
(b) \[\left( - 4, \frac{8}{3} \right)\]
(c) \[\left( 4, - \frac{8}{3} \right)\]
(d) none of these
The slope of the tangent to the curve x = t2 + 3t − 8, y = 2t2 − 2t − 5 at the point (2, −1) is _____________ .
\[\frac{22}{7}\]
\[\frac{6}{7}\]
\[\frac{7}{6}\]
\[- \frac{6}{7}\]
The line y = mx + 1 is a tangent to the curve y2 = 4x, if the value of m is ________________ .
1
2
3
`1/2`
The normal at the point (1, 1) on the curve 2y + x2 = 3 is _____________ .
x + y = 0
x − y = 0
x + y + 1 = 0
x − y = 1
The normal to the curve x2 = 4y passing through (1, 2) is _____________ .
x + y = 3
x − y = 3
x + y = 1
x − y = 1
none of these
Chapter 16: Tangents and Normals

RD Sharma solutions for Class 12 Maths chapter 16 - Tangents and Normals
RD Sharma solutions for Class 12 Maths chapter 16 (Tangents and Normals) include all questions with solution and detail explanation. This will clear students doubts about any question and improve application skills while preparing for board exams. The detailed, step-by-step solutions will help you understand the concepts better and clear your confusions, if any. Shaalaa.com has the CBSE Class 12 Maths solutions in a manner that help students grasp basic concepts better and faster.
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Concepts covered in Class 12 Maths chapter 16 Tangents and Normals are Maximum and Minimum Values of a Function in a Closed Interval, Maxima and Minima, Simple Problems on Applications of Derivatives, Graph of Maxima and Minima, Approximations, Tangents and Normals, Increasing and Decreasing Functions, Rate of Change of Bodies Or Quantities, Introduction to Applications of Derivatives.
Using RD Sharma Class 12 solutions Tangents and Normals exercise by students are an easy way to prepare for the exams, as they involve solutions arranged chapter-wise also page wise. The questions involved in RD Sharma Solutions are important questions that can be asked in the final exam. Maximum students of CBSE Class 12 prefer RD Sharma Textbook Solutions to score more in exam.
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