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Q. The vertices of a hyperbola H are (±6,0) and its eccentricity is 52. Let N be the normal to H at a point in the first quadrant and parallel to the line 2x+y=22. If d is the length of the line segment of N between H and the y-axis then d2 is equal to

JEE MainJEE Main 2023Conic Sections

Solution:

image
H:x236y29=1
equation of normal is 6xcosθ+3ycotθ=45
 slope =2sinθ=2
θ=π4
Equation of normal is 2x+y=15
P:(asecθ,btanθ)
P(62,3) and K(0,15)
d2=216