Find the The Slopes of the tangent and the normal to the following curves at the indicated points :

Find the The Slopes of the tangent and the normal to the following curves at the indicated points :

at x = 9

Find the The Slopes of the tangent and the normal to the following curves at the indicated points :

y = x³ – x at x = 2

Find the The Slopes of the tangent and the normal to the following curves at the indicated points :

y = 2x² + 3 sin x at x = 0

Find the The Slopes of the tangent and the normal to the following curves at the indicated points :

x = a (θ – sin θ), y = a(1 + cos θ) at

θ = – π/2

Find the The Slopes of the tangent and the normal to the following curves at the indicated points :

x = a cos³ θ, y = a sin³ θ at θ = π/4

Find the The Slopes of the tangent and the normal to the following curves at the indicated points :

x = a(θ – sin θ), y = a(1 – cos θ) at θ = π/2

Find the The Slopes of the tangent and the normal to the following curves at the indicated points :

y = (sin 2x + cot x + 2)² at x = π/2

Find the The Slopes of the tangent and the normal to the following curves at the indicated points :

x² + 3y + y² = 5 at (1, 1)

Find the The Slopes of the tangent and the normal to the following curves at the indicated points :

xy = 6 at (1, 6)

Find the values of a and b if the 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 a point on the curve y = x³ – 2x² – 2x at which the tangent lines are parallel to the line y = 2x – 3.

Find a point on the curve y² = 2x³ at which the Slope of the tangent is 3

Find a point on the curve xy + 4 = 0 at which the tangents are inclined at an angle of 45^o with the x–axis.

Find a point on the curve y = x² where the Slope of the tangent is equal to the x – coordinate of the point.

At what point 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^o with the x–axis?

Find a point 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 a point on the curve y = 3x² + 4 at which the tangent is perpendicular to the line whose slope is

Find the point on the curve x² + y² = 13, the tangent at each one of which is parallel to the line 2x + 3y = 7.

Find the point on the curve 2a²y = x³ – 3ax² where the tangent is parallel to the x – axis.

At what points on the curve y = x² – 4x + 5 is the tangent perpendicular to the line 2y + x = 7?

Find the point on the curve at which the tangents are parallel to the

x – axis

Find the point on the curve at which the tangents are parallel to the y – axis.

Find the point on the curve x² + y² – 2x – 3 = 0 at which the tangents are parallel to the x – axis

Find the point on the curve x² + y² – 2x – 3 = 0 at which the tangents are parallel to the y – axis.

Find the point on the curve at which the tangents are parallel to x – axis

Find the point on the curve at which the tangents are parallel to y – axis

Show that the tangents to the curve y = 7x³ + 11 at the points x = 2 and x = – 2 are parallel.

Find the point on the curve y = x³ where the Slope of the tangent is equal to x – coordinate of the point.

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 curves at the indicated points:

y = x⁴ – 6x³ + 13x² – 10x + 5 at (0, 5)

Find the equation of the tangent and the normal to the following curves at the indicated points:

y = x⁴ – 6x³ + 13x² – 10x + 5 at x = 1 y = 3

Find the equation of the tangent and the normal to the following curves at the indicated points:

y = x² at (0, 0)

Find the equation of the tangent and the normal to the following curves at the indicated points:

y = 2x² – 3x – 1 at (1, – 2)

Find the equation of the tangent and the normal to the following curves at the indicated points:

at (2, – 2)

Find the equation of the tangent and the normal to the following curves at the indicated points:

y = x² + 4x + 1 at x = 3

Find the equation of the tangent and the normal to the following curves at the indicated points:

at (a cos θ, b sin θ)

Find the equation of the tangent and the normal to the following curves at the indicated points:

at (a sec θ, b tan θ)

Find the equation of the tangent and the normal to the following curves at the indicated points:

y² = 4a x at (a/m², 2a/m)

Find the equation of the tangent and the normal to the following curves at the indicated points:

Find the equation of the tangent and the normal to the following curves at the indicated points:

xy = c² at (ct, c/t)

Find the equation of the tangent and the normal to the following curves at the indicated points:

at (x₁, y₁)

Find the equation of the tangent and the normal to the following curves at the indicated points:

at (x₀, y₀)

Find the equation of the tangent and the normal to the following curves at the indicated points:

x^2/3 + y^2/3 = 2 at (1, 1)

Find the equation of the tangent and the normal to the following curves at the indicated points:

x² = 4y at (2, 1)

Find the equation of the tangent and the normal to the following curves at the indicated points:

y² = 4x at (1, 2)

Find the equation of the tangent and the normal to the following curves at the indicated points:

4x² + 9y² = 36 at (3 cos θ, 2 sin θ)

Find the equation of the tangent and the normal to the following curves at the indicated points:

y² = 4ax at (x₁, y₁)

Find the equation of the tangent and the normal to the following curves at the indicated points:

at

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 curves at the indicated points:

x = θ + sin θ, y = 1 + cos θ at θ = π/2.

Find the equation of the tangent and the normal to the following curves at the indicated points:

at t = 1/2

Find the equation of the tangent and the normal to the following curves at the indicated points:

x = at², y = 2at at t = 1.

Find the equation of the tangent and the normal to the following curves at the indicated points:

x = a sec t, y = b tan t at t.

Find the equation of the tangent and the normal to the following curves at the indicated points:

x = a (θ + sin θ), y = a (1 – cos θ) at θ

Find the equation of the tangent and the normal to the following curves at the indicated points:

x = 3 cos θ – cos³ θ, y = 3 sin θ – 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 the 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 is

perpendicular to the line 5y – 15x = 13.

Find the equation of all lines having slope 2 and that are tangent to the curve

Find the equation of all lines of slope zero and that is tangent to the curve

Find the equation of the tangent to the curve which is parallel to the line 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 touches the straight line for all at the point (a, b).

Find the equation of the tangent to the curve x = sin 3t, y = cos 2t at

At what points will be tangents to the curve y = 2x³ – 15x² + 36x – 21 be parallel to the 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 to intersection of the following curves :

y² = x and x² = y

Find the angle to intersection of the following curves :

y = x² and x² + y² = 20

Find the angle to intersection of the following curves :

2y² = x³ and y² = 32x

Find the angle to intersection of the following curves :

x² + y² – 4x – 1 = 0 and x² + y² – 2y – 9 = 0

Find the angle to intersection of the following curves :

Find the angle to intersection of the following curves :

x² + 4y² = 8 and x² – 2y² = 2

Find the angle to intersection of the following curves :

x² = 27y and y² = 8x

Find the angle to intersection of the following curves :

x² + y² = 2x and y² = x

Find the angle to intersection of the following curves :

y = 4 –x² and y = x²

Show that the following set of curves intersect orthogonally :

y = x³ and 6y = 7 – x²

Show that the following set of curves intersect orthogonally :

x³ – 3xy² = – 2 and 3x² y – y³ = 2

Show that the following set of curves intersect orthogonally :

x² + 4y² = 8 and x² – 2y² = 4.

Show that the following curves intersect orthogonally at the indicated points :

x² = 4y and 4y + x² = 8 at (2, 1)

Show that the following curves intersect orthogonally at the indicated points :

x² = y and x³ + 6y = 7 at (1, 1)

Show that the following curves intersect orthogonally at the indicated points :

y² = 8x and 2x² + y² = 10 at (1, 2√2)

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 curves to interest orthogonally.

and xy = c²

Find the condition for the following set of curves to interest orthogonally.

and

Show that the curves and interest at right angles

If the straight line xcosα + ysinα = p touches the curve then prove that

a²cos²α–b²sin²α = ρ².

The equation to the normal to the curve y = sinx at (0, 0) is

The equation of the normal to the curve y = x + sin x cos x at 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 = at², y = 2at 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 on 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 = k is

The equation 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 + 3t – 8, y = 2t² – 2t – 5 at point (2, –1) is

At what points 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 = acos³ θ, y = a sin³θ at the point is

If the curves y = 2 e^x and y = ae^–x interest orthogonally, then a =

The point on the curve y = 6x – x² at which the tangent to the curve is inclined at 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 sin²x and y = cos² x at 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

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

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.

Write the value of 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 coordinate axes, then write the value of

If the tangent line at a point (x, y) on the curve y = f(x) is parallel to y-axis, find the value of

Find the slope of the normal at the point ‘t’ on the curve

Write the coordinates of the point on the curve y² = x where the tangent line makes an angle with x-axis.

Write the angle made by the tangent to the curve x = e^t cos t, y = e^t sin t atwith the x-axis.

Write the equation of the normal to the curve y = x + sin x cos x at

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 of 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 = e^x at their point of intersection.

Write the slope of the normal to the curve at the point

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 = cosx at (0, 1).

Write the equation of the tangent drawn to the curve y = sinx at the point (0, 0).