A point P is taken on the circle x2+y2=a2.PN,PM are drawn perpendicular to axes. The locus of the pole of line MN is
A
x2+y2=a2
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B
x−2+y−2=a−2
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C
x−2−y−2=a−2
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D
x−2+y−2=a−4
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Solution
The correct option is Cx−2+y−2=a−2 M≡(0,asinθ)andN≡(acosθ,0) ∴ Let A(h,k) is pole of line MN. Then, chord of contact from A(h,k) to circle x2+y2=a2 is hx+ky=a2 ..... (1) And equation of MN is y=−sinθcosθ(x−acosθ) ycosθ+sinθx=acosθsinθ ...... (2) Equations (1) and (2) represent the same line, then kcosθ=hsinθ=a2asinθcosθ ∴k=acosecθ and h=asecθ ∴ Equation of locus of A(h,k) is a2h2+a2k2=1 Replace h→x and k→y ⇒x−2+y−2=a−2