Given: Equation of the curve ……….(i) Slope of the tangent at point is ……….(ii) Slope of the line is ……….(iii) From eq. (ii) and (iii), From eq. (i), Therefore, required point is (1,...
Choose the correct answer in: The slope of the normal to the curve y = 2x^ 2 + 3 sin x at x = 0 is (A) 3 (B) 1/ 3 (C) –3 (D)- 1/ 3
Given: Equation of the curve ……….(i) Slope of the tangent at point is Slope of the tangent at (say) Slope of the normal = Therefore,The correct option is option (D)...
Find the equation of the tangent to the curve (fig 1)which is parallel to the line 4x- 2y+ 5= 0
fig 1: SOLUTION: Given: Equation of the curve ……….(i) Slope of the tangent at point is = ……….(ii) Again slope of the line is ……….(iii) According to the question, [Parallel lines have same...
Find the equations of the tangent and normal to the hyperbola x^2/a^2 + y^2/ b^2 =1 at the point (x0 , y0 ).
Given: Equation of the hyperbola ……….(i) ……….(ii) Slope of tangent at is Equation of the tangent at is ……….(iii) Since lies on the hyperbola (i), therefore, From eq....
Prove that the curves x = y ^2 and xy = k cut at right angles if 8k ^2 = 1.
Given: Equations of the curves are …..(i) and ……….(ii) Substituting the value of in eq. (ii), we get Putting the value of in eq. (i), we get Therefore, the point of intersection is...
Find the equations of the tangent and normal to the parabola y^ 2 = 4ax at the point (at^2 , 2at).
Given: Equation of the parabola ……….(i) Slope of the tangent at = Slope of the tangent at the point = Slope of the normal = Equation of the tangent at the point = And Equation...
Find the equation of the normals to the curve y = x ^3 + 2x + 6 which are parallel to the line x + 14y + 4 = 0.
Equation of the curve ….(i) Slope of the tangent at = Slope of the normal to the curve at = ……….(ii) But Slope of the normal (given) = = From eq. (i), at at Therefore, the...
Find the equation of the normal at the point (am^2 ,am^3 ) for the curve ay^2 = x ^3
Equation of the curve ……….(i) Slope of the tangent at the point = = Slope of the normal at the point = Equation of the normal at = ...
Find the points on the curve x ^2 + y ^2 – 2x – 3 = 0 at which the tangents are parallel to the x-axis.
Equation of the curve ……….(i) [tangent is parallel to axis] From eq. (i), Therefore, the required points are (1, 2) and
For the curve y = 4x^ 3 – 2x ^5 , find all the points at which the tangent passes through the origin.
Given: Equation of the curve ……….(i) Slope of the tangent at passing through origin (0, 0) = = Substituting this value of in eq. (i), we get, or or From eq. (i), at From eq....
Find the points on the curve y = x ^3 at which the slope of the tangent is equal to the y-coordinate of the point.
Given: Equation of the curve ………(i) Slope of tangent at = ……….(ii) According to question, Slope of the tangent = coordinate of the point or or From eq. (i), at The point is (0,...
Show that the tangents to the curve y = 7x ^3 + 11 at the points where x = 2 and x = – 2 are parallel.
Equation of the curve Slope of tangent at = At the point Slope of the tangent = At the point Slope of the tangent = Since, the slopes of the two tangents are equal. Therefore, tangents...
Find the equation of the tangent line to the curve y = x^ 2 – 2x +7 which is (a) parallel to the line 2x – y + 9 = 0 (b) perpendicular to the line 5y – 15x = 13
Given: Equation of the curve ……….(i) Slope of tangent = …….(ii) (a) Slope of the line is Slope of tangent parallel to this line is also = 2 From eq. (ii), From eq. (i), Therefore,...
Find the equations of the tangent and normal to the given curves at the indicated points:(v) x = cost, y = sin t at t= π/4
Equation of the curves are and Slope of the tangent at = (say) Slope of the normal at is Point = = = Equation of the tangent is And Equation of the normal is ...
Find the equations of the tangent and normal to the given curves at the indicated points: (iii) y = x ^3 at (1, 1) (iv) y = x ^2 at (0, 0)
Equation of the curve ……….(i) Now value of at (1, 1) At = (say) Slope of the normal at (1, 1) is Equation of the tangent at (1, 1) is And Equation of the normal at (1, 1) is ...
Find the equations of the tangent and normal to the given curves at the indicated points: (i) y = x ^4 – 6x^ 3+ 13x^ 2 –10x + 5 at (0, 5) (ii) y = x ^4 – 6x ^3 + 13x ^2 – 10x + 5 at (1, 3)
(i) Equation of the curve Now value of at (0, 5) At (say) Slope of the normal at (0, 5) is Equation of the tangent at (0, 5) is And Equation of the normal at (0, 5) is ...
Find points on the curve x^ 2/9+ y^2/16=1 at which the tangents are (i) parallel to x-axis (ii) parallel to y-axis
Given: Equation of the curve ……….(i) ……….(ii) (i) If tangent is parallel to axis, then Slope of tangent = 0 = 0 From eq. (i), Therefore, the points on curve (i) where...
Find the equations of all lines having slope 0 which are tangent to the curve y= 1/( x^2-2x+3)
Given: Equation of the curve ……….(i) = = But according to question, slope = 0 = 0 From eq. (i), Therefore, the point on the curve which tangent has slope 0 is Equation of the tangent...
Find the equation of all lines having slope 2 which are tangents to the curve y=1 /(3-x) , x ≠ 3
Given: Equation of the curve = = Slope of the tangent at But according to question, slope = = 2 which is not possible. Hence, there is no tangent to the given curve having slope...
Find the equation of all lines having slope – 1 that are tangents to the curve y= 1 /(1-x) , x ≠ 1.
Given: Equation of the curve ……….(i) = = Slope of the tangent at But according to question, slope = = or From eq. (i), when And when Points of contact are (2, 1) and And...
Find the point on the curve y = x^ 3 – 11x + 5 at which the tangent is y = x – 11.
Given: Equation of the curve ……….(i) Equation of the tangent ……….(ii) From eq. (i), = Slope of the tangent at But from eq. (ii), the slope of tangent = From eq. (i), when And when ...
Find a point on the curve y = (x – 2)^2 at which the tangent is parallel to the chord joining the points (2, 0) and (4, 4).
Let the given points are A (2, 0) and B (4, 4). Slope of the chord AB = Equation of the curve is Slope of the tangent at = If the tangent is parallel to the chord AB, then Slope of tangent =...
Find points at which the tangent to the curve y = x ^3 – 3x^ 2 – 9x + 7 is parallel to the x-axis.
Given: Equation of the curve ……….(i) Since, the tangent is parallel to the axis, i.e., From eq. (i), when when Therefore, the required points...
Find the slope of the normal to the curve x=1-asinθ, y=bcos^2θ at θ=π/2
Given: Equations of the curves are and and and = Slope of the tangent at = And Slope of the normal at = =
Find the slope of the normal to the curve x=acos^3thetha, y=asin^3thetha at thetha=pi/4
Given: Equations of the curves are and = and and = Slope of the tangent at = And Slope of the normal at = = 1
Find the slope of tangent to the curve y= x^3-3x+2 at the given point whose coordinate is 3.
Given: Equation of the curve ……….(i) Slope of the tangent at point to the curve (i) = = 27 – 3 = 24
Find the slope of tangent to the curve y=x^3-x+1 at the given point whose x-coordinate is 2.
Given: Equation of the curve ……….(i) Slope of the tangent at point to the curve (i) = = 12 – 1 = 11
Find the slope of tangent to the curve y=(x-1)/(x+1),x not equal to 0 at x=10
Given: Equation of the curve ……….(i) = = ……….(ii) Slope of the tangent at point to the curve (i) = =
Find the slope of tangent to the curve y=3x^4-4x at x=4
Equation of the curve ……….(i) Slope of the tangent to the curve = Value of at the point Slope of the tangent at point to the curve (i) = = 768 – 4 = 764