Question

If the equation ax² + 2hxy + by² + 2gx + 2fy = 0 has one line as the bisector of the angle between co-ordinate axes, then ______.

Options

• (a+ b)² = 4(h² + t²)

• (a+ b)² = 4(h² + g² + t²)

• (a+ b)² = 4h²

• (a+ b)² = 4(h² + g²)

Answer 1

If the equation ax² + 2hxy + by² + 2gx + 2fy = 0 has one line as the bisector of the angle between co-ordinate axes, then (a+ b)² = 4h².

Explanation:

Given, equation of line

ax² + 2hxy + by² + 2gx + 2fy = 0

Let the slope of line is m₁ and m₂

∴ `"m"_1 + "m"_2 = (- "2h")/"b"`

`"m"_1"m"_2 = "a"/"b"`

Given, one line is bisector of angle between coordinate axes.

∴ m₁ = tan`pi/4` = 1

⇒ 1 + m₂ = - 2h/b    ...(i)

`"m"_2 = "a"/"b"`   ...(ii)

From Eqs. (i) and (ii), we get

`1 + "a"/"b" = (- 2"h")/"b"`

⇒ a + b = - 2h

(a + b)² = 4h²