Find $p$ and $k$ if the equation $p x^{2} - 8 x y + 3 y^{2} + 14 x + 2 y + k = 0$ represents a pair of perpendicular lines.

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Solution

$\begin{matrix} P x^{2} - 8 x y + 3 y^{2} + 14 x + 2 y + k = 0 C o m p a r i n g w i t h \Rightarrow a x^{2} + 2 h x y + b y^{2} + 2 g x = 2 f y + c = 0 \Rightarrow a = p, h = - 4, b = 3, g = 7, f = 1, c = k f e q u a t i o n o f l i n e s r e p r e s e n t p e r p e n d i c u l a r \Rightarrow a + b = 0 \Rightarrow p + 3 = 0 ∴ p = - 3 A l s o, ∣ ∣ ∣ ∣ \begin{matrix} a h g h b f g f c \end{matrix} ∣ ∣ ∣ ∣ = 0 \Rightarrow∴ ∣ ∣ ∣ ∣ \begin{matrix} - 3 - 47 - 431 71 k \end{matrix} ∣ ∣ ∣ ∣ = 0 \Rightarrow - 9 k + 3 - 16 k - 28 - 175 = 0 \Rightarrow - 25 k = 200 ∴ k = - 8 A n s . \end{matrix}$

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Find p and k if the equation px2−8xy+3y2+14x+2y+k=0 represents a pair of perpendicular lines.

Find $p$ and $k$ if the equation $p x^{2} - 8 x y + 3 y^{2} + 14 x + 2 y + k = 0$ represents a pair of perpendicular lines.