Let L_1, L_2 be the lines passing through the point P(0,1) and touching the parabola 9 x^2…

Question

Let $L_1, L_2$ be the lines passing through the point $P(0,1)$ and touching the parabola $9 x^2+12 x+18 y-14=0$. Let $Q$ and $R$ be the points on the lines $L_1$ and $L_2$ such that the $$\triangle P Q R$$ is an isosceles tria...

Answer: 68

Accepted Answer

$\begin{gathered} 9{x}^{2}+12x+18y-14=0 \ {(x+\frac{2}{3})}^{2}=-2(y-1)\ldots (1) \end{gathered}$

Equation of tangent to (1)

$\begin{gathered} t(x+\frac{2}{3})=-(y-1)+\frac{1}{2}{t}^{2}(0,1) \ \Rightarrow \frac{2}{3}t=\frac{1}{2}{t}^{2}\Rightarrow t=0,\frac{4}{3}\Rightarrow m=0,m=-\frac{4}{3} \end{gathered}$

$\begin{gathered} \Rightarrow |\frac{m+\frac{4}{3}}{1-\frac{4}{3}m}|=|m|\Rightarrow |\frac{3m+4}{3-4m}|=|m| \ \Rightarrow 3m+4=m(3-4m)3m+4=-m(3-4m) \ 3m+4=3m-4{m}^{2}3m+4=-3m+4{m}^{2} \ 4{m}^{2}+4=04{m}^{2}-6m-4=0 \ {m}_{1}+{m}_{2}=\frac{3}{2},{m}_{1}{m}_{2}=-1 \ \Rightarrow {m}_{2}^{1}+{m}_{2}^{2}=\frac{17}{4} \ 16({m}_{2}^{1}+{m}_{2}^{2})=68 \end{gathered}$

Solutions · 1