Questions
The locus of the foot of the perpendicular from the center of the hyperbola xy = 1 on a variable tangent is
a
x2−y22=4xy
b
x2+y22=2xy
c
x2+y2=4xy
d
x2+y22=4xy
detailed solution
Correct option is D
Let the foot of perpendicular from O(0,0) to the tangent to the hyperbola be P(h, k). Slope of OP=khThen the equation of tangent to the hyperbola is y−k=−hk(x−h)or hx+ky=h2+k2Solving it with xy = 1, we have hx+kx=h2+k2or hx2−h2+k2x+k=0This equation must have real and equal roots. HenceD=0or h2+k22−4hk=0or x2+y22=4xyTalk to our academic expert!
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If the distance between two directrices of a rectangular hyperbola is 15, then the distance between its foci in units is:
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