Tangents & Normals — Slope, equations
Applications of Derivatives
15
JEE Qs
8%
Hard
75
min
Master differentiation techniques and always evaluate the derivative at the exact point of tangency to find the correct slope.
🧮 Key Formulas
✅ Key Points for JEE
- 1The derivative dy/dx evaluated at a specific point (x1, y1) on a curve gives the slope of the tangent line to the curve at that point.
- 2The normal line to a curve at a point is perpendicular to the tangent line at the same point. Their slopes thus satisfy m_t * m_n = -1.
- 3When solving problems, always ensure you differentiate the given function correctly and substitute the point of tangency (x1, y1) into dy/dx to get the specific slope.
- 4For curves defined parametrically (x=f(t), y=g(t)), the slope of the tangent is dy/dx = (dy/dt) / (dx/dt). For implicitly defined curves, use implicit differentiation.
- 5Horizontal tangents occur where dy/dx = 0, and vertical tangents occur where dx/dy = 0 (or dy/dx is undefined).
⚠️ Common Mistakes
- ✕Failing to substitute the coordinates of the point of tangency into dy/dx after differentiation, leading to a general slope expression instead of a numerical value.
- ✕Incorrectly calculating the slope of the normal as 1/m_t instead of -1/m_t, or failing to handle cases where m_t = 0 (horizontal tangent, vertical normal) or m_t is undefined (vertical tangent, horizontal normal).
- ✕Using a point other than the point of tangency (x1, y1) in the point-slope form (y - y1 = m(x - x1)) for the equation of the line.
📝 Practice Questions
See allQ10.Let the function f(x) = (x2 + 1) x2 −ax + 2 + cos |x| be not differentiable at the two points x = α = 2 and x = β . Then the distance of the point (α, β) from the line 12x + 5y + 10 = 0 is equal to : (1) 5 (2) 4 (3) 3 (4) 2
Q24.Let the function, f(x) = {−3ax2a2 + bx,−2, xx <⩾11 be differentiable for all x ∈R, where a > 1, b ∈R. If the area of the region enclosed by y = f(x) and the line y = −20 is α + β√3, α, β ∈Z , then the value of α + β is ________
Q8. Let f(x) = ∫x20 t2−8t+15et dt, respectively, are : (1) 2 and 3 (2) 2 and 2 (3) 3 and 2 (4) 1 and 3
Q23.If the set of all values of a, for which the equation 5x3 −15x −a = 0 has three distinct real roots, is the interval (α, β), then β −2α is equal to ______
Q13.A spherical chocolate ball has a layer of ice-cream of uniform thickness around it. When the thickness of the ice-cream layer is 1 cm , the ice-cream melts at the rate of 81 cm3/min and the thickness of the ice-cream layer decreases at the rate of 1 cm/min. The surface area (in cm2 ) of the chocolate ball (without the ice- 4π cream layer) is : (1) 196π (2) 256π (3) 225π (4) 128π
Q20.Let →a = ^i + 2^j + 3^k,→b = 3^i + ^j −^k and →c be three vectors such that →c is coplanar with →a and →b. If the vector →C is perpendicular to →b and →a ⋅→c = 5, then |→c| is equal to (1) √116 (2) 3√21 (3) 16 (4) 18
NCERT Chapters
- Class 11 Maths Ch 10: Straight Lines
- Class 12 Maths Ch 5: Continuity and Differentiability
- Class 12 Maths Ch 6: Applications of Derivatives