सिद्ध कीजिये कि वक्र `a_(1)x^(2)+b_(1)y^(2)=1` तथा `a_(2)x^(2)+b_(2)y^(2)=1` एक-दूसरे को समकोण पर काटेंगे |
यदि `(1)/(a_(1))-(1)/(b_(1))=(1)/(a_(2))-(1)/(b_(2))`

Explanation

प्रदत्त वक्र `a_(1)x^(2)+b_(1)y^(2)=1" "…(1)`
तथा `" "a_(2)x^(2)+b_(2)y^(2)=1" "…(2)`
माना वक्रों (1) व (2) का प्रतिच्छेद बिन्दु `(x_(1),y_(1))` है
समीकरण (1) से, `2a_(1)x+2b_(1)y(dy)/(dx)=0`
`implies" "(dy)/(dx)=-(a_(1))/(b_(1))(x)/(y)`
`:." "((dy)/(dx))_((x_(1)","y_(1)))=-((a_(1)x_(1))/(b_(1)y_(1)))=m_(2)" "("माना")" "...(3)`
समीकरण (2) से,
`2a_(2)x+2b_(2)y(dy)/(dx)=0`
`:." "((dy)/(dx))_((x_(1)","y_(1)))=(-(a_(2)x_(1))/(b_(2)y_(1)))=m_(2)" "("माना")" "...(4)`
`because` दोनों वक्र परस्पर समकोण पर प्रतिच्छेद होते हैं |
अत: `" "m_(1)m_(2)=-1`
`implies" "(-(a_(1)x_(1))/(b_(1)y_(1)))(-(a_(2)x_(1))/(b_(2)y_(1)))=-1`
`implies" "(x_(1)^(2))/(y_(1)^(2))=-(b_(1)b_(2))/(a_(1)a_(2))" "...(5)`
समीकरण (2) को समीकरण (1) में से घटाने पर,
`(a_(1)-a_(2))x^(2)+(b_(1)-b_(2))y^(2)=0`
`implies" "(x^(2))/(y^(2))=-((b_(1)-b_(2))/(a_(1)-a_(2)))`
`:." "(x_(1
)^(2))/(y_(1)^(2))=-((b_(1)-b_(2))/(a_(1)-a_(2)))" "...(6)`
समीकरण (5) व (6) से,
`-(b_(1)b_(2))/(a_(1)a_(2))=-((b_(1)-b_(2)))/((a_(1)-a_(2)))`
`implies" "((a_(1)-a_(2)))/(a_(1)a_(2))=((b_(1)-b_(2)))/(b_(1)b_(2))`
`implies" "(1)/(a_(2))-(1)/(a_(1))=(1)/(b_(2))-(1)/(b_(1))`
`implies" "(1)/(a_(1))-(1)/(a_(2))=(1)/(b_(1))-(1)/(b_(2))`