SIR PLEASE HELP ME WITH THIS QUESTIONS :

1. Find the equation of the tangent to the circle x^2+y^2=16 drawn from the point (1,4).

2. How many triangles can be formed by joining any 3 of the 9 points when

i) no 3 of them  are collinear?

ii) 5 of them are collinear?

3. AB is a line of fixed length, 6 units, joining the points A(t,0) and B which lies on thepositive y axis. P is a point on AB distant 2 units from A. Express the coordinates of B and P in terms of t. Find the locus of P as t varies.

4. In a triangle ABC,

(b^2 - c^2) / (b^2 + c^2) = sin(B - C) / sin(B + C), prove that it is either a right-angled or isoceles triangle.

Asked by kandappan | 8th Feb, 2020, 11:29: AM

Expert Answer:

1.
Equation of circle in the standard form is (x-h)2 + (y-k)2 = r2
Given equation of circle is x2 + y2 = 16
rightwards double arrow h equals 0 comma space k equals 0 space a n d space r equals 4
E q u a t i o n space o f space tan g e n t space t o space t h e space c i r c l e space w i t h space s l o p e space m space i s
y minus k equals m left parenthesis x minus k right parenthesis plus-or-minus square root of m squared a squared plus a squared end root
rightwards double arrow y equals m x plus-or-minus square root of 16 m squared plus 16 end root space.... space left parenthesis i i right parenthesis
T h e space a b o v e space e q u a t i o n space o f space tan g e n t space p a s s e s space t h r o u g h space t h e space p o i n t space left parenthesis 1 comma space 4 right parenthesis
rightwards double arrow 4 equals m plus-or-minus square root of 16 m squared plus 16 end root
rightwards double arrow open parentheses 4 minus m close parentheses squared equals 16 m squared plus 16
rightwards double arrow 16 plus m squared minus 8 m equals 16 m squared plus 16
rightwards double arrow 15 m squared plus 8 m equals 0
rightwards double arrow m equals 0 space space o r space space m equals negative 8 over 15
F r o m space left parenthesis i i right parenthesis comma space w e space h a v e
y equals 4 space space o r space space y equals negative fraction numerator 8 x over denominator 15 end fraction plus-or-minus square root of 16 cross times 64 over 225 plus 16 end root equals negative fraction numerator 8 x over denominator 15 end fraction plus-or-minus fraction numerator square root of 16 cross times 289 end root over denominator 15 end fraction equals negative fraction numerator 8 x plus-or-minus 68 over denominator 15 end fraction
rightwards double arrow y equals 4 space o r space 15 y equals negative 8 x plus 68 space space o r space space minus 8 x minus 68
H e n c e comma space t h e space e q u a t i o n space o f space tan g e n t space i s space y equals 4 space o r space 15 y equals negative 8 x plus 68 space o r space y equals negative 8 x minus 68.

Answered by Renu Varma | 10th Feb, 2020, 11:24: AM