Math Problems: Circle (tangent line) and Functions?

Question

1.) A parabola passes through the points (0,2) and (-1,5), the latter being its vertex. Given the function, f(x)=ax^2 + bx +c, find the values of a,b and c.

2.) Find the points on the circle x^2 -2x +y^2 = 24 such that the tangent line at those points are parallel to the line y= (3/4)x

Answer 1

q1

f(0) = 0 + 0 + c = 2

c = 2

f(x) = a(x+1)^2 + 5

= ax^2 + 2ax + a + 5

a + 5 = 2

a = -3

b = 2a = -6

answer : a = -3, b = -6, c = 2

q2

x^2 - 2x + y^2 = 24

differentiating,

2x - 2 + 2y(dy/dx) = 0

dy/dx = (2 - 2x) / 2y

= (1 - x ) / y = 3 / 4

y = 4(1-x)/3

plugging in the above into the original eq.,

x^2 - 2x + 16(1-x)^2 /9 = 24

x^2 (1 + 16/9) + x(-2 - 2*16/9) + 16/9 = 24

25x^2 - 50x + 16 = 9*24

25x^2 - 50x - 200 = 0

25(x - 4)(x + 2) = 0

x = -2, 4

answer : points (-2,4) and (4,-4).