Question

A line passing through the points (a,2a) and (−2,3) is perpendicular to the line 4x+3y+5=0 , find the value of a.

Answer 1

The value of 'a' in the given question is 18/5.

`\[ m_1 = (2a - 3)/(a + 2) \]\[ m_2 = -(4)/(3) \]`

The lines are perpendicular, so
`\( m_1 \cdot m_2 = -1 \)`.

Now, substitute the expressions for
`\( m_1 \)` and
`\( m_2 \)` into the perpendicularity condition:

`\[ (2a - 3)/(a + 2) \cdot \left(-(4)/(3)\right) = -1 \]`

Multiply both sides by
`\(-(3)/(4)\)` to simplify:

`\[ (2a - 3)/(a + 2) = (3)/(4) \]`

Cross-multiply to eliminate fractions:

4(2a - 3) = 3(a + 2)

Expand and solve for a:

8a - 12 = 3a + 6

Combine like terms:

5a = 18

Solve for a:

`\[ a = (18)/(5) \]`

So, the correct value for a is
`\( (18)/(5) \)`.