Solution
Step 1: Find the coordinates of vertex A.
Vertex A is the point of intersection of the sides AB and AC.
-
px+qy=1 (Equation of AB)
-
qx+py=1 (Equation of AC)
Subtracting (2) from (1):
Assuming p=q:
Substitute x=y into equation (1):
So, the vertex is A=(p+q1,p+q1).
Step 2: Identify the midpoint of the base BC.
The problem states that the base BC is bisected at the point (p,q). By definition, the point that bisects the base is the midpoint M.
Step 3: Find the equation of the median AM.
A median is a line segment joining a vertex to the midpoint of the opposite side. We need the equation of the line passing through A(p+q1,p+q1) and M(p,q).
Using the two-point form: y−y1=x2−x1y2−y1(x−x1)
The slope (m) of the median AM is:
m=p+qp(p+q)−1p+qq(p+q)−1
The equation of the line is:
y−q=(p2+pq−1pq+q2−1)(x−p)
(y−q)(p2+pq−1)=(x−p)(pq+q2−1)
Step 4: Simplify the equation.
Expanding both sides:
y(p2+pq−1)−q(p2+pq−1)=x(pq+q2−1)−p(pq+q2−1)
y(p2+pq−1)−p2q−pq2+q=x(pq+q2−1)−p2q−pq2+p
Cancel common terms (−p2q−pq2) from both sides:
y(p2+pq−1)+q=x(pq+q2−1)+p
x(pq+q2−1)−y(p2+pq−1)+(p−q)=0
Final Answer:
The equation of the median through vertex A is:
(pq+q2−1)x−(pq+p2−1)y+(p−q)=0
