Find the Equation of tangent which passes through Point $(0, 1)$ and line touches touches the Parabola $9 x^{2} + 12 x + 18 y - 14 = 0.$

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Solution

$\begin{matrix} 9 x^{2} + 12 x + 18 y - 14 = 0 e q u a t i o n o f tan g e n t w i t h s l o p e m t o p a r a b o l a a n d p a s s e s t h r o u g h (0, 1) y - 1 = m (x - 0) \Rightarrow y - 1 = m x ∴ y = m x + 1 s o l v e tan g e n t a n d p a r a b o l a 9 x^{2} + 12 x + 18 (m x + 1) - 14 = 0 \Rightarrow 9 x^{2} + 12 x + 18 m x + 18 - 14 = 0 \Rightarrow 9 x^{2} + (12 + 18 m) x + 4 = 0 i . e l i n e t o u c h i . e e q u a l r o o t s D = 0 b^{2} - 4 a c = 0 {(12 + 18 m)}^{2} - 4 \times 9 \times 4 = 0 36 {(2 + 3 m)}^{2} - 4 \times 36 = 0 4 + 9 m^{2} + 12 m - 4 = 0 9 m^{2} + 12 m = 0 ∴ m (9 m + 12) = 0 ∴ m = 0, m = \frac{- 12}{9} = \frac{- 4}{3} \end{matrix}$

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