RD Sharma solutions for Mathematics [English] Class 12 chapter 16 - Tangents and Normals [Latest edition]

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 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 ?

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 ?

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 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 ?

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 log_e 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) ?

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  \[\left( 1, 2\sqrt{2} \right)\] ?

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 a²cos^{2 \[\alpha\]} \[-\] b²sin² \[\alpha\] = p² ?

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 = e^t cos t, y = e^t 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 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 tangent drawn to the curve \[y = \sin x\] at the point (0,0) ?

The equation to the normal to the curve y = sin x at (0, 0) is ___________ .

The equation of the normal to the curve y = x + sin x cos x at x = `π/2` is ___________ .

The equation of the normal to the curve y = x(2 − x) at the point (2, 0) is ________________ .

The point on the curve y² = x where tangent makes 45° angle with x-axis is ______________ .

If the tangent to the curve x = a t², y = 2 at is perpendicular to x-axis, then its point of contact is _____________ .

The point on the curve y = x² − 3x + 2 where tangent is perpendicular to y = x is ________________ .

The point on the curve y² = x where tangent makes 45° angle with x-axis is ____________________ .

The point at the curve y = 12x − x² where the slope of the tangent is zero will be _____________ .

The angle between the curves y² = x and x² = y at (1, 1) is ______________ .

The equation of the normal to the curve 3x² − y² = 8 which is parallel to x + 3y = 8 is ____________ .

The equations of tangent at those points where the curve y = x² − 3x + 2 meets x-axis are _______________ .

The slope of the tangent to the curve x = t² + 3 t − 8, y = 2t² − 2t − 5 at point (2, −1) is ________________ .

At what point the slope of the tangent to the curve x² + y² − 2x − 3 = 0 is zero

The angle of intersection of the curves xy = a² and x² − y² = 2a² is ______________ .

If the curve ay + x² = 7 and x³ = y cut orthogonally at (1, 1), then a is equal to _____________ .

If the line y = x touches the curve y = x² + bx + c at a point (1, 1) then _____________ .

The slope of the tangent to the curve x = 3t² + 1, y = t³ −1 at x = 1 is ___________ .

The curves y = ae^x and y = be^−x cut orthogonally, if ___________ .

The equation of the normal to the curve x = a cos³ θ, y = a sin³ θ at the point θ = π/4 is __________ .

If the curves y = 2 e^x and y = ae^−x intersect orthogonally, then a = _____________ .

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 __________ .

The angle of intersection of the parabolas y² = 4 ax and x² = 4ay at the origin is ____________ .

The angle of intersection of the curves y = 2 sin² x and y = cos 2 x at \[x = \frac{\pi}{6}\] is ____________ .

Any tangent to the curve y = 2x⁷ + 3x + 5 __________________ .

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 _____________ .

The line y = mx + 1 is a tangent to the curve y² = 4x, if the value of m is ________________ .

The normal at the point (1, 1) on the curve 2y + x² = 3 is _____________ .

The normal to the curve x² = 4y passing through (1, 2) is _____________ .