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238. EXAMPLE If the normals at three points P, Q and R in a s be the focus, meet point O and prove that SP. SQ. SR a SO As in the previous question we know that the normals a the points (ami, am 1), (am am2) and (am3, 2am3) meet in the point (h k), if m1 1m2 1m3 0 2a -h m2m3 m3m1 m1m2 (2) |
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Answer» `y=mx-2am-am^3` `(am_1^2,-2am_1)(am_2^2,-2am_2)(am_3^2,-2am_3)` `am^3+2am-mh+k=0` `m_1+m_2+m_3=0` `m_1m_2+m_1m_3+m_2m_3=(2a-h)/a` `m_1m_2m_3=-k/a` =SP*SQ*SR `=(am_1^2+a)(am_2^2+a)(am_3^2+a)` `=a^3(m_1^2+1)(m_2^2+!)(m_3^2+1)` `=a^3(m_1^2*m_2^2m_3^2+m_1^2m_2^2+m_1^2m_3^2+n_1^2+m_3^2m_3^2+m_2+m_3^2+1)` `=a^3[(h^2-2ah+a^2)/a^2]` `a^3/a^2(h-a)^2` `a(OS)^2`. |
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