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-

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