Ceist leudachaidh
Tha an loidhne \(l\) na tansaint ris a' chearcall le meadhan \({C_1}\) agus an co-aontar:
\({x^2} + {y^2} - 4x - 6y + 8 = 0\)
Tha na co-chomharran \((1,5)\) aig puing suathaidh A.
Question
a) Seall gur e \(l\) an co-aontar aig an loidhne \(2y - x = 9\)
\({x^2} + {y^2} - 4x - 6y + 8 = 0\)
Meadhan: \({C_1}(2,3)\)
Airson \({C_1}(2,3)\) agus \(A(1,5)\)
\({m_{CA}} = \frac{{5 - 3}}{{1 - 2}} = \frac{2}{{ - 1}} = - 2\)
\(\Rightarrow {m_1} = \frac{1}{2}\)
'S e a' phuing air an tansaint \(A(1,5)\)
agus 's e an caisead \(= \frac{1}{2}\)
Mar sin 's e an co-aontar:
\(y - 5 = \frac{1}{2}(x - 1)\)
\(2y - 10 = x - 1\)
\(2y - x = 9\)
Tha an co-aontar:
\({x^2} + {y^2} + 2x + 2y - 18 = 0\)
aig a' chearcall le meadhan \({C_2}\)
Question
b) Seall gu bheil an loidhne \(l\) cuideachd na tansaint ris a' chearcall.
Airson trasnadh loidhne agus cearcall fuasgail:
\(2y - x = 9\)
\({x^2} + {y^2} + 2x + 2y - 18 = 0\)
Ionadaich \(x = 2y - 9\) do cho-aontar a' chearcaill.
\({(2y - 9)^2} + {y^2} + 2(2y - 9) + 2y - 18 = 0\)
\(4{y^2}-36y+81+{y^2}+4y-18+2y-18=0\)
\(5{y^2} - 30y + 45 = 0\)
\(5({y^2} - 6y + 9) = 0\)
\(5(y - 3)(y - 3) = 0\)
\(y = 3\)
Agus bhon nach eil ann ach aon fhuasgladh, tha an loidhne na tansaint ris a' chearcall.
Dh'fhaodadh cuideachd a bhith air a shealltainn leis an discriminant.
Question
c) Mas e B a' phuing-suathaidh, obraich a-mach faid AB.
Nuair a tha \(y = 3\)
\(x = 2 \times 3 - 9 = - 3\)
agus mar sin 's e a' phuing-suathaidh \(B( - 3,3)\)
Airson \(A(1,5)\) agus \(B( - 3,3)\)
\(AB = \sqrt {1 - {{( - 3)}^2} + {{(5 - 3)}^2}}\)
\(= \sqrt {{4^2} + {2^2}} = \sqrt {20} = 2\sqrt 5\)
