CONTOH SOAL 4 IRISAN KERUCUT

 $\begin{array}{ll}\\ 7.&\textbf{EBTANAS SMA IPA 1994}\\ &\textrm{Persamaan garis singgung pada elips} \\ &3x^{\displaystyle 2}+4y^{\displaystyle 2}-12=0\quad  \textrm{yang tegak lurus pada}\\ &\textrm{garis}\quad y=x+1\quad\textrm{ adalah}\:....\\ &\begin{array}{llll}\\ \textrm{A}.&\displaystyle y=x+\sqrt{7}\quad \textrm{atau}\quad y=x-\sqrt{7}\\ \textrm{B}.&\displaystyle y=-x+\sqrt{7}\quad \textrm{atau}\quad y=-x-\sqrt{7}\\ \textrm{C}.&\displaystyle y=x+5\quad \textrm{atau}\quad y=x-5\\ \textrm{D}.&\displaystyle y=-x+5\quad \textrm{atau}\quad y=-x-5\\ \textrm{E}.&\displaystyle y=-x+2\sqrt{5}\quad \textrm{atau}\quad y=-x-2\sqrt{5}\end{array}\\\\ &\textrm{Jawab}:\quad \textbf{B}\\ &\textrm{Perhatikan bahwa garis singgung elips }\\ &\left( 3x^{\displaystyle 2}+4y^{\displaystyle 2}-12=0\Leftrightarrow \displaystyle \frac{x^{\displaystyle 2}}{4}+\frac{y^{\displaystyle 2}}{3}=1 \right)\\ &y=x+1\quad \textrm{yang bergradien}\:\: m=1,\quad \textrm{maka}\\ &\textrm{gradien garis yang tegak lurus adalah}\:\: m=-1\\ &\textrm{dan rumus garis singgungnya adalah}\\ &y=mx\pm \sqrt{a^{\displaystyle 2}m^{\displaystyle 2}+b^{\displaystyle 2}}\Rightarrow y=-x\pm \sqrt{4.(-1)^{\displaystyle 2}+3}\\ &\Leftrightarrow y=-x\pm \sqrt{7}  \end{array}$.

$\begin{array}{ll}\\ 8.&\textbf{EBTANAS SMA IPA 1994}\\ &\textrm{Persamaan asimtot hiperbola dengan persamaan} \\ &16y^{\displaystyle 2}-9x^{\displaystyle 2}-36=0\quad\textrm{ adalah}\:....\\ &\begin{array}{llll}\\ \textrm{A}.&\displaystyle 9x+16y=0\quad \textrm{dan}\quad 9x-16y=0\\ \textrm{B}.&\displaystyle 3x+2y=0\quad \textrm{dan}\quad 3x-2y=0\\ \textrm{C}.&\displaystyle 2x+3y=0\quad \textrm{dan}\quad 2x-3y=0\\ \textrm{D}.&\displaystyle 4x+3y=0\quad \textrm{dan}\quad 4x-3y=0\\ \textrm{E}.&\displaystyle 3x+4y=0\quad \textrm{dan}\quad 3x-4y=0\end{array}\\\\ &\textrm{Jawab}:\quad \textbf{E}\\ &\textrm{Diketahui bahwa  persamaan hiperbola}\\ &16y^{\displaystyle 2}-9x^{\displaystyle 2}-36=0\Leftrightarrow \displaystyle \frac{y^{\displaystyle 2}}{\left( \displaystyle \frac{36}{16} \right)^{\displaystyle 2}}-\frac{x^{2}}{2^{\displaystyle 2}}-1=0\\ &\Leftrightarrow \displaystyle \frac{y^{2}}{\left( \displaystyle \frac{3}{2} \right)^{\displaystyle 2}}-\frac{x^{\displaystyle 2}}{2^{\displaystyle 2}}=1,\quad \textrm{maka}\quad a=\displaystyle \frac{3}{2}\quad \textrm{dan}\quad b=2\\ &\textrm{Sedangkan rumus asimtotnya adalah}\\ &y=\displaystyle \pm \frac{a}{b}x\Rightarrow y=\displaystyle \pm \frac{\left( \displaystyle \frac{3}{2} \right)}{2}\Leftrightarrow y=\displaystyle \pm \frac{3}{4}x\\ &\Leftrightarrow 3x+4y=0\quad \textrm{atau}\quad 3x-4y=0  \end{array}$.