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Question

Let a,b,c be real numbers with a2+b2+c2=1 Show that the equation ∣ ∣axbycbx+aycx+abx+ayax+byccy+bcx+acy+baxby+c∣ ∣ = 0
represents a straight line.

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Solution

C1aC1Δ=1a∣ ∣ ∣a2xabyacbx+aycx+aabx+a2yax+byccy+bacx+a2cy+baxby+c∣ ∣ ∣Applying C1C1+bC2+cC3Δ=1a∣ ∣ ∣(a2+b2+c2)xay+bxcx+a(a2+b2+c2)ybycaxcy+b(a2+b2+c2)b+cyaxby+c∣ ∣ ∣Δ=1a∣ ∣xay+bxcx+aybycaxb+cy1b+cycaxby∣ ∣,as a2+b2+c2=1C2C2bC1 and C3C3cC1then Δ=1a∣ ∣xayaycaxb1cyaxby∣ ∣
=1ax∣ ∣ ∣x2axyaxycaxb1cyaxby∣ ∣ ∣R1R1+yR2+R3Δ=1ax∣ ∣ ∣x2+y2+100ycaxb1cyaxby∣ ∣ ∣
On expanding along R1Δ=(x2+y2+1)axax(ax+by+c)=(x2+y2+1)(ax+by+c)Given Δ=0ax+by+c=0, which represents a straight line.
[x2+y2+10,being +ve].

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