Question

1 point

Find the y - coordinate of the point of intersection of straight lines represented by (1) and (2), given the following equations:

ax + by + c = E ---- (1)

�
�
+
�
�
+
�
2
bx+cy+d
2
= F ---- (2)


Given that

�
=
�
=
0
E=F=0

Arithmetic mean of a and b is c. Geometric mean of a and b is d. Choose the correct option. Note:

Arithmetic mean of m and n is
�
+
�
2
2
m+n
​


Geometric mean of m and n is
�
�
mn
​


(
2
�
2
�
−
�
�
−
�
2
2
�
2
−
�
2
−
�
�
)
(
2b
2
−a
2
−ab
2a
2
b−ab−b
2

​
)

(
�
2
�
−
�
−
1
)
(
a−b
a
2

​
−1)

(
2
�
2
�
−
�
�
−
�
2
2
�
2
−
�
2
−
�
�
)
(
2b
2
−b
2
−ab
2b
2
b−ab−b
2

​
)

(
�
2
�
−
�
−
1
)
(
a−b
b
2

​
−1)

Answer 1

The geometric mean of a and b is d.

Explanation:

To find the y-coordinate of the point of intersection of the two lines, we need to solve the system of equations formed by (1) and (2).

Given the equations:

(1) ax + by + c = 0

(2) bx + cy + d = 0

We are also given the conditions: E = F = 0.

To solve for the point of intersection, we can eliminate one variable (either x or y) by multiplying one equation by a suitable constant to make the coefficients of either x or y equal in magnitude but opposite in sign.

Let's eliminate x by multiplying equation (1) by b and equation (2) by -a:

b(ax + by + c) = 0

-a(bx + cy + d) = 0

Simplifying, we get:

abx + b^2y + bc = 0

-abx - acy - ad = 0

Adding these two equations together, we have:

(b^2 - ab)x + (bc - ac)y + (bc - ad) = 0

Since E = F = 0, we can conclude that (bc - ad) = 0. This condition implies that either b = 0 or c = 0.

If b = 0, then the line represented by (1) is a vertical line. In this case, we cannot find the point of intersection as it does not exist.

Therefore, the correct option is:

The geometric mean of a and b is d.