Question

(4,2), (k, - 3) are conjugate points with respect to the circle x2 + y2 - 5x + 8y + 6 = 0.​

Answer 1

EXPLANATION.

=> ( 4,2) , ( k, -3) are the conjugate point with

respect the circle = x² + y² - 5x + 8y + 6 = 0

$\sf : \implies \: equation \: of \: circle \: = {x}^{2} + {y}^{2} - 5x + 8y + 6 = 0 \\ \\ \sf : \implies \: its \: polar \: point \: = (4,2)$

$\sf : \implies \: chord \: of \: contact \: \\ \\ \sf : \implies \: t \: = 0 \: \: \: or \: \: s_{1} = 0 \\ \\ \sf : \implies \: 4x + 2y - 5( \frac{x + 4}{2}) + 8( \frac{y + 2}{2}) + 6 = 0 \\ \\ \sf : \implies \: 4x + 2y - \frac{5x}{2} - 10 + 4y + 8 + 6 = 0 \\ \\ \sf : \implies \: 4x - \frac{5x}{2} + 6y + 4 = 0 \\ \\ \sf : \implies \frac{8x - 5x + 12y + 8}{2} = 0 \\ \\ \sf : \implies \: 3x + 12y + 8 = 0 \: \: \: ......(1)$

$\sf : \implies \: put \: the \: points \: (k \: , - 3) \: \: in \: \: equation \: (1) \\ \\ \sf : \implies \: 3k + 12( - 3) + 8 = 0 \\ \\ \sf : \implies \: 3k - 36 + 8 = 0 \\ \\ \sf : \implies \: k \: = \frac{28}{3}$

$\sf : \implies \: \green{{ \underline{value \: of \: k \: = \dfrac{28}{3} }}}$

Answer 2

$\underline{\underline{\blue{Given:-}}}$

=> ( 4,2) , ( k, -3) are the conjugate point with

respect the circle = x² + y² - 5x + 8y + 6 = 0

$\underline{\underline{\red{Solution:-}}}$

$\begin{gathered}\sf \implies \: equation \: of \: circle \: = {x}^{2} + {y}^{2} - 5x + 8y + 6 = 0 \\ \\ \sf \implies \: its \: polar \: point \: = (4,2)\end{gathered}$$\begin{gathered}\sf \implies \: chord \: of \: contact \: \\ \\ \sf \implies \: t \: = 0 \: \: \: or \: \: s_{1} = 0 \\ \\ \sf \implies \: 4x + 2y - 5( \frac{x + 4}{2}) + 8( \frac{y + 2}{2}) + 6 = 0 \\ \\ \sf \implies \: 4x + 2y - \frac{5x}{2} - 10 + 4y + 8 + 6 = 0 \\ \\ \sf \implies \: 4x - \frac{5x}{2} + 6y + 4 = 0 \\ \\ \sf \implies \frac{8x - 5x + 12y + 8}{2} = 0 \\ \\ \sf : \implies \: 3x + 12y + 8 = 0 \: \: \: ......(1)\end{gathered}$$\begin{gathered}\sf \implies \: put \: the \: points \: (k \: , - 3) \: \: in \: \: equation \: (1) \\ \\ \sf \implies \: 3k + 12( - 3) + 8 = 0 \\ \\ \sf \implies \: 3k - 36 + 8 = 0 \\ \\ \sf \implies \: k \: = \frac{28}{3}\end{gathered}$

$\sf \implies \: \pink{\boxed{ \underline{value \: of \: k \: = \dfrac{28}{3} }}}$

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