सिद्ध कीजिए कि वक्रों `x=3 cos t-cos^(3) t` तथा `y=3 sin t-sin^(3) t` के बिन्दु t पर अभिलम्ब का समीकरण `4(y cos^(3) t-x sin^(3) t)=3 sin 4t` हैं |

Explanation

दिये गये वक्रों के समीकरण
`x=3 cos t-cos^(3)t`
तथा `" "y=3 sin t-sin^(3) t`
अब, t के सापेक्ष अवकलन करने पर
`(dx)/(dt)=3(d)/(dt)(cos t)-(d)/(dt)(cos^(3) t)`
तथा `" "(dy)/(dt)=3(d)/(dt)(sin t)-(d)/(dt)(sin^(3) t)`
`implies" "(dx)/(dt)=3xx(-sin t)-3 cos^(2)t(d)/(dt)(cos t)`
तथा `" "(dy)/(dt)=3xx(cos t)-3 sin^(2)t(d)/(dt)(sin t)`
`implies" "(dx)/(dt)=-3sint-3cos^(2)txx(-sint)`
तथा `" "(dy)/(dt)=3cost-3sin^(2)txx(cost)`
`implies" "(dx)/(dt)=-3sintxxsin^(2)t`
तथा `" "(dy)/(dt)=3cost(1-sin^(2)t)`
`implies" "(dx)/(dt)=-3 sintxxsin^(2)t`
तथा `" "(dy)/(dt)=3 costxxcos^(2)t`
`implies" "(dx)/(dt)=-3sin^(3)t" तथा"(dy)/(dt)=3cos^(3)t`
अब, `" "(dy)/(dx)=(dy//dt)/(dx//dt)=(3cos^(3)t)/(-3sin^(3)t)=-cot^(3)t`
`:.` अभिलम्ब का समीकरण
`y-(3sint-sin^(3)t)=(1)/(-cot^(3)t)[x-(3cost-cos^(3)t)]`
`[becausey-y_(1)=-(1)/(dy//dx)(x-x_(1))]`
`(y-3sint+sin^(3)t)=(sin^(3)t)/(cos^(3)t)(x-3cost+cos^(3)t)`
`implies" "ycos^(3)t-3sintcos^(3)t+sin^(3)tcos^(3)t`
`=xsin^(3)t-3sin^(3)tcost+sin^(3)tcos^(3)t`
`implies" "ycos^(3)t-xsin^(3)t=3sintcos^(3)t-3sin^(3)t cos t`
`implies" "ycos^(3)t-xsin^(3)t=3sint cost(cos^(2)t-sin^(2)t)`
`=3 sint cost(cos2t)xx(2)/(2)" "[becausecos^(2)t-sin^(2)t=cos2t]`
`=(3)/(2)sin2txxcos2t" "[because2sintcost=sin2t]`
`=(3)/(2)sin2txxcos2txx(2)/(2)`
`=(3)/(4)sin4t" "[because2sin2tcos2t=sin4t]`
`:.4(ycos^(3)t-xsin^(3)t)=3sin4t`