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$The locus of the midpoints of the chord of the circle, x^{2} + y^{2} = 25 which is tangent to the$ $hyperbola, \frac{x^{2}}{9} - \frac{y^{2}}{16} = 1 is$

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a

x2+y22−16x2+9y2=0

b

x2+y22−9x2−16y2=0

c

x2+y22−9x2+144y2=0

d

x2+y22−9x2+16y2=0

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detailed solution

Correct option is D

Equation of the chord having mid point h,k to the circle x2+y2=25 is hx+ky=h2+k2it can be written as y=−hkx+h2+k2kGiven this is tangent to the hyperbola x29−y216=1c2=a2m2-b2⇒h2+k2k2=9h2k2−16⇒h2+k22=9h2−16k2The locus is x2+y22=9x2-16y2 or x2+y22-9x2+16y2=0

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