If the line x + y + 1 = 0 touches the circle $x^{2} + y^{2} - 3 x + 7 y + 14 = 0$ . Find points of contact

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Solution

$x + y + 1 = 0 \Rightarrow y = - 1 - x$
The line touches the circle $x^{2} + y^{2} - 3 x + 7 y + 14 = 0$
$\Rightarrow x^{2} + {(- 1 - x)}^{2} - 3 x + 7 (- 1 - x) + 14 = 0$
$\Rightarrow x^{2} + 1 + x^{2} + 2 x - 3 x - 7 - 7 x + 14 = 0$
$\Rightarrow 2 x^{2} - 8 x + 8 = 0$
$\Rightarrow x^{2} - 4 x + 4 = 0$
$\Rightarrow {(x - 2)}^{2} = 0$
$\Rightarrow x - 2 = 0$
$\Rightarrow x = 2$
Put $x = 2$ in $y = - x - 1 = - 2 - 1 = - 3$
Hence the point of contact is $(2, - 3)$

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If the line x + y + 1 = 0 touches the circle x2+y2−3x+7y+14=0. Find points of contact

x+y+1=0⇒y=−1−xThe line touches the circle x2+y2−3x+7y+14=0⇒x2+(−1−x)2−3x+7(−1−x)+14=0⇒x2+1+x2+2x−3x−7−7x+14=0⇒2x2−8x+8=0⇒x2−4x+4=0⇒(x−2)2=0⇒x−2=0⇒x=2Put x=2 in y=−x−1=−2−1=−3Hence the point of contact is (2,−3)

If the line x + y + 1 = 0 touches the circle $x^{2} + y^{2} - 3 x + 7 y + 14 = 0$ . Find points of contact