Let C be the circle with the center (2,3) and radius 5 a) show that P(5,7) lies on C and find the equation of the tangent at P b) show that the line 3x−4y+31=0 is a tangent to C


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Question Number 199834 by Calculusboy last updated on 10/Nov/23

$L e t C b e t h e c i r c l e w i t h t h e c e n t e r (2, 3) a n d r a d i u s 5$ $a) s h o w t h a t P (5, 7) l i e s o n C a n d f i n d t h e$ $e q u a t i o n o f t h e t a n g e n t a t P$ $b) s h o w t h a t t h e l i n e 3 x - 4 y + 31 = 0 i s a t a n g e n t t o C$
Commented by Calculusboy last updated on 10/Nov/23

$o k a y s i r$
	Answered by cortano12 last updated on 10/Nov/23

	$(a) {(5 - 2)}^{2} + {(7 - 3)}^{2} = 9 + 16 = 25 = 5^{2} = R^{2}$ $so that P (5, 7) lies on circle$ $(b) equation of tangent at P (5, 7)$ $is (5 - 2) (x - 2) + (7 - 3) (y - 3) = 25$ $3 x - 6 + 4 y - 12 = 25$ $3 x + 4 y = 43$ $(c) the line 3 x - 4 y + 31 = 0$ $touching the circle so$ $∣ \frac{3.2 - 4.3 + 31}{5} ∣= \frac{25}{5} = 5 = R$
	Commented by Calculusboy last updated on 10/Nov/23

	$t h a n k s s i r$
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