Miscellaneous

Chapter 6 Class 12 Application of Derivatives
Serial order wise

### Transcript

Question 3 Show that the normal at any point ΞΈ to the curve x = a cos π + a π sin π, y = a sin π β a π cos π is at a constant distance from the origin.Given curve π=π cosβ‘π+π π sinβ‘π , π=π sinβ‘πβ π π cosβ‘π We need to show distance of a normal from (0, 0) is constant First , calculating Equation of Normal We know that Slope of tangent is ππ¦/ππ₯ ππ/ππ= (ππ/ππ½)/(ππ/ππ½) Finding ππ/ππ½ Given π₯=π cosβ‘π+π π sinβ‘π Diff. w.r.t ΞΈ ππ₯/ππ= π(π cosβ‘π + π π sinβ‘π )/ππ ππ₯/ππ= (π(π cosβ‘π))/ππ + (π(ππ sinβ‘π))/ππ ππ₯/ππ = π (βsinβ‘π )+π (π(π sinβ‘π))/ππ Using product rule (u v)β = uβ v + vβ u ππ₯/ππ=β a sin ΞΈ+a (ππ/ππ sin ΞΈ+ (π(π ππ π))/ππ ΞΈ) ππ₯/ππ=β a sin ΞΈ+a ( sin ΞΈ+ΞΈ cosβ‘γΞΈ γ ) ππ₯/ππ=β a sin ΞΈ+a sin ΞΈ+π ΞΈ cosβ‘γΞΈ γ ππ/ππ½=π π½ πππβ‘γπ½ γ Finding ππ/ππ½ Given π¦=π π ππβ‘πβπ π πππ β‘π Diff. w.r.t ΞΈ ππ¦/ππ= π(π π ππβ‘πβπ π πππ β‘π)/ππ ππ¦/ππ=a cos ΞΈβa (ππ/ππ cos ΞΈ+ (π(πππ  π))/ππ ΞΈ) ππ¦/ππ=a cos ΞΈβa ( cos ΞΈβΞΈ sinβ‘γΞΈ γ ) ππ¦/ππ=a cos ΞΈβa cos ΞΈ+π ΞΈ sinβ‘γΞΈ γ ππ/ππ½=π π½ πππβ‘γπ½ γ Now, ππ/ππ= (ππβππ½)/(ππβππ½) ππ¦/ππ₯=(π π sinβ‘π)/(π π cosβ‘π ) ππ¦/ππ₯=sinβ‘π/cosβ‘π ππ/ππ= πππβ‘π½ We know that Slope of tangent of Γ Slope of normal = β1 tan ΞΈ Γ Slope of normal = β1 Slope of normal = (β1)/tanβ‘π Slope of normal =βπππβ‘π½ Equation of normal which passes through the curve π = a cos ΞΈ + a ΞΈ sin ΞΈ & π = a sin ΞΈ β a ΞΈ cos ΞΈ & has slope βπππβ‘π½ is We know that Equation of line passing through (π₯1 , π¦1) & having slope m is (π¦βπ¦1) = m(π₯βπ₯1) (π¦β(π sinβ‘πβπ cosβ‘π ))=βππ¨π­β‘π½(π₯β(π cosβ‘π+π π sinβ‘π )) (π¦βπ sinβ‘π+π π cosβ‘π )=(βπππβ‘π½)/πππβ‘π½ (π₯βπ cosβ‘π+π π sinβ‘π ) π¬π’π§β‘π½(π¦βπ sinβ‘π+π π cosβ‘π )=βππ¨π¬β‘π½(π₯βπ cosβ‘πβπ π sinβ‘π ) π¦ sinβ‘πβπ sin2 π+π π .cosβ‘π sinβ‘π=βπ₯ cosβ‘π+π cos2 π+π πsinβ‘π cosβ‘π π¦ sinβ‘πβπ sin2 π+π₯ cosβ‘πβπ cos2 π=π π sinβ‘π cosβ‘πβπ π sinβ‘π cosβ‘π π¦ sinβ‘π+π₯ cosβ‘πβπ sin2 πβπ cos2 π=0 π¦ sinβ‘π+ π₯ cosβ‘πβπ (ππππ π½+ππππ π½)=0 π¦ sinβ‘π+π₯ cosβ‘πβπ (π)=0 π πππβ‘π½+π πππβ‘π½ βπ = π We know that Distance of line ax + by + c = 0 from point (x1, y1) is d = |πππ + πππ +π|/β(π¨^π + π©^π ) Finding Distance of Normal from Origin Equation of normal is π πππβ‘π½+π πππβ‘π½ βπ = π Comparing with aπ₯ + bπ¦ + c = 0 β΄ a = cos ΞΈ, b = sin ΞΈ & c = β a We need to find distance of normal from origin i.e. ππ = 0 & ππ = 0 π= |cosβ‘γπ(0) + sinβ‘γπ(0) β πγ γ |/β(cos^2β‘π + sin^2β‘π ) d = |0 + 0 β π|/β1 d = |βπ|/1 d = π/1 d = a d = Constant Hence, distance of normal from origin is a constant. Hence proved.

#### Davneet Singh

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 14 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.