Find-the-equation-of-the-line-which-is-tangent-to-the-parabola-y-2-12x-and-forms-an-angle-of-45-with-the-line-y-3x-4-

Question Number 183241 by depressiveshrek last updated on 23/Dec/22

F i n d t h e e q u a t i o n o f t h e l i n e w h i c h

i s t a n g e n t t o t h e p a r a b o l a y^{2} = 12 x

a n d f o r m s a n a n g l e o f 45 ° w i t h

t h e l i n e y = 3 x - 4 .

Answered by cortano1 last updated on 24/Dec/22

L e t t h e l i n e i s y = m x + c

a n d 1 =∣ \frac{m - 3}{1 + 3 m} ∣ \Rightarrow∣ m - 3 ∣=∣ 3 m + 1 ∣

\Rightarrow (4 m - 2) (2 m + 4) = 0; m = - 2; \frac{1}{2}

t h e l i n e w h i c h t a n g e n t t o y^{2} = 12 x

\Rightarrow {(m x + c)}^{2} = 12 x

\Rightarrow m^{2} x^{2} + (2 m c - 12) x + c^{2} = 0, Δ = 0

\Rightarrow {(2 m c - 12)}^{2} - {(2 m c)}^{2} = 0

\Rightarrow (4 m c - 12) (- 12) = 0

\Rightarrow c = \frac{3}{m} = - \frac{3}{2}; 6

∴ {\begin{cases} y = - 2 x - \frac{3}{2} \\ y = \frac{1}{2} x + 6 \end{cases}

Commented by cortano1 last updated on 24/Dec/22