Mathematics

Problem

Let

A(2,-3)

and

B(6,5)

be two vertices of a triangle

ABC

. The point

C(x,y)

satisfies the following conditions: (a) the midpoint of

AC

lies on the line

x - y + 1 = 0

, and (b) the line

BC

is perpendicular to

5x + 2y - 7 = 0

. a. (i) Obtain the coordinates of the midpoint of

AC

in terms of

x

and

y

, and hence write down the first equation relating

x

and

y

. b. (ii) Using condition (b), derive the second equation relating

x

and

y

. c. (iii) Solve the simultaneous equations to find the coordinates of

C

. d. (iv) Verify that the midpoint found in part (i) indeed satisfies the given line equation.