IIT JEE , AIEEE Forum - Mathematics , Analytical Geometry

rphy

the locus of the mid pts of a chord of x² + y² =4 which subtends a right angle at the origin is?

also do the conditions of tangency remain d same for a conjugate hyberbola as it is for a hyperbola?

tarun007

for Q1
let the points be (h,k)
using T=S1 we get the equation of the chord as:
hx+ky=h^2+k^2
use the principle of homogenization :
x^2+y^2 - 4((hx+ky)/(h^2+k^2))^2=0

now this is the combined equation of the two lines subtending 90 degree angle on origin.

now put the condition of perpendicularity of the two lines. i.e.
coeff of x^2 +coeff of y^2=0.

you will get the following eqn:

h^2+k^2=2.

replace (h,k) by (x,y).

A MUCH MUCH SHORTER METHOD IS :(first draw the diagram)
using simple geometry (and simple trigo) you will see that every such required point will be at a distance of under root 2 from the origin. Hence the answer: x^2+y^2=2

otherwise refer to the above solution.

Please rate me if it was helpful !!!!!!

joyfrancis

it is clear from the figure that distance between the center and the origin is root2 units.

Apply distance formula

h^2+k^2=2

Locus

x^2+y^2=2

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