Problem

Find the intervals in which the function $f$ give…

Problem 5 Easy Difficulty

Show that the normal at any point $\theta$ to the curve
$x=a \cos \theta+a \theta \sin \theta, y=a \sin \theta-a \theta \cos \theta$
is at a constant distance from the origin.

Answer

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Calculus 1 / AB

NCERT Class 12 Part 1 - Math

Chapter 6

Application of Derivatives

Section 6

Maxima and Minima

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Video Transcript

this question we have to show that the normal at any point data to recall X equals to a cost data Plus 80 design data. Why it was to assign data minus 80 decosta is at a constant distance from the origin. Right? So these are the paramedic evacuation of the girl. Like. So the approach to solve this question is first will find the slope of the tangent at a point X ray. We'll find with the help of that. We'll find the slope of the normal late and we'll find the aggression of the normal. We'll find the distance of the normal from the Khoisan. Right? And we'll do that. It is questioned or not. Right. So to find the slope of the tangent we have to find the X. By data and divide by the data. So the expiry date A. That would be equal to minuses science data. Bless 80 decosta plus. Yes I wanted to Right we just close to 80. The cost leader when we differentiate y with respect to cater. Very good. A cost later minus 80 to 70 to plus a course to the right. Sorry. Um, yeah. Yeah. A cost a less data. Scientific, a scientist to Minour a costly delayed, which is equal to 80 design data. So the slope of retention would be equal today by by the X. Which can be written as day by by the theater upon the expiry date data. Right? So this is the close to 10 data. So the slope of the tangent stand data, the slope of the normal Would be equal to -1 x 10 data. Or we can say minor scott. Right. So we can write the equation of the normal at any point. Like so at any point the equation of the normal would divide minus uh minus the coordinate that is like ordinary. That is assigned to the minor state. Acosta, mm hmm. Is equal to the slope which is minus car stayed up on science data, multiplied by x minus x coordinate of the point. That is a Costa plus data. Science data. They Yeah. Mhm. So when we simplify this, we get the equation of the normal is X cost data less white sign data is equal to a. So the distance of the normal from the origin judo judo would be equal to geo plus models of zero plus zero minus a, divided by the square root of courses. Square data plus sine squared theta which is equal to a. Which is a constant. Like hence it is gold. That the distance of the normal from the origin will be a question.

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