Q.
If the line y−2=0 is the directrix of the parabola x2−ky+32=0,k=0 and the parabola intersects the circle x2+y2=8 at two real distinct points, then the absolute value of k is
3062
191
NTA AbhyasNTA Abhyas 2020Conic Sections
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Answer: 16
Solution:
x2−ky+32=0 ⇒x2=k(y−k32)
Put, x=X,y−k32=Y
The equation of directrix is Y+4k=0
i.e. y−k32+4k=0
But, y−2=0 is the directrix. ⇒k32−4k=2 ⇒k2+8k−128=0 ⇒k=−16 or k=8
For k=8, the parabola is x2=8(y−4) which does not intersect the circle.
For k=−16, the parabola is x2=−16(y+2) which intersects the circle at two real distinct points. ⇒ Absolute value of k=∣−16∣=16