Trigonometrical Ellipse

Geometry Level 4

For the ellipse 2 x 2 − 2 x y + 4 y 2 − ( 3 + 2 ) = 0 2x^2-2xy+4y^2 - (3+\sqrt 2) = 0 , Find the inclination of it's major axis with the x axis in a coordinate plane.

The Answer is in the form of 180 a \dfrac {180}{a} degrees where a is an integer. What is the value of a?

4 0 8 6 2 10

Vineet Golcha by vineet golcha , 22, India

This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try refreshing the page, (b) enabling javascript if it is disabled on your browser and, finally, (c) loading the non-javascript version of this page . We're sorry about the hassle.

2 solutions

Prakhar Bindal
Nov 3, 2015

Partially Differentiating The Equation Of Ellipse Once With respect to x and then wrt y and solving those

equations we get that ellipse is centred at origin . we assume a variable chord of the ellipse y= mx (passing

through the centre. Any general point on the line can be taken as (rcosx , rsinx) where x is the inclination of

the line with the x axis . now for the major axis r must be maximum (ellipse is a central curve every chord

passing through its centre is bisected at the centre) . therefore r = length of semi .- major axis if the given

point lies on the ellipse. substituting this in equation of ellipse and finding an expression for r in terms of x

we obtain that the maxima is obtained when x= 22.5 degrees

PS.- We Know That The Problem Is Not Original :P :)

4 Helpful 0 Interesting 0 Brilliant 0 Confused

every problem cant be original ! it was an interesting one so I shared it .

vineet golcha - 5 years, 7 months ago

In this case you can explicitly find r as a function of x and then optimise it.But if there would also have been terms containing x and y then it would not be possible to express r as a function of x as it would become a quadratic equation in r with coefficients functions of x.Then how would you maximise/minimise r ?

Indraneel Mukhopadhyaya - 4 years, 3 months ago

What does partial differential signify here ?

Vishal Yadav - 4 years, 2 months ago

Clearly , this is a rotated ellipse (perhaps maybe even shifted but we are not interested in finding that).Suppose that if we rotate the coordinate axes by k radians ,the ellipse has its major and minor axes parallel to the coordinate axes.The transformation is thus : x=Xcos(k) +Ysin(k) and y= -Xsin(k) + Ycos(k).Substitute the transformed coordinates in the equation of ellipse to get the equation of ellipse in rotated frame.By equating coefficient of XY=0 , we get tan(2k) = 1 from which we see that k=π/8 radians or k=180/8 degrees.So, the answer is 8.

1 Helpful 0 Interesting 0 Brilliant 0 Confused

0 pending reports

×

Problem Loading...

Note Loading...

Set Loading...