Question

the angle between lines 5x+3y=7 and 3x-5y=15​

Answer 1

Let us know the rule for finding the angle between two lines first:

We consider two lines

• y = m₁x + c₁
• y = m₂x + c₂

If θ be the angle between them, we can write

θ = tan⁻¹ ( |m₁ - m₂| ) / ( 1 + m₁ m₂ )

Now we move to answer the given question:

The given straight lines are

• 5x + 3y = 7 or, y = (- 5/3) x + 7/3 ..... (i)
• 3x - 5y = 15 or, y = (3/5) x - 15/5 ..... (ii)

Here m₁ = - 5/3, m₂ = 3/5

Using the formula, we get the angle between the lines (i) and (ii):

θ = tan⁻¹ ( | - 5/3 - 3/5 | ) / { 1 + (- 1) }

= tan⁻¹ ( | - 5/3 - 3/5 | ) / 0

= tan⁻¹ ( ∞ )

= 90°

So the angle between the given lines is 90°.

There is another way to find the solution:

We have already found that,

• m₁ = - 5/3
• m₂ = 3/5

∴ m₁ m₂ = (- 5/3) × 3/5

= - 1

This shows that the lines (i) and (ii) are perpendicular to each other.