Question

If the slopes of the lines given by the equation ax² + 2hxy + by² = 0 are in the ratio 5 : 3, then the ratio h² : ab = ______.

Options

• 3 : 5

• 16 : 15

• 5 : 3

• 15 : 16

Answer 1

If the slopes of the lines given by the equation ax² + 2hxy + by² = 0 are in the ratio 5 : 3, then the ratio h² : ab = 16 : 15.

Explanation:

Given line,

ax² + 2hxy + by² = 0

∴ m₁ + m₂ = `- "2h"/"b"`

m₁m₂ = `"a"/"b"`

`because ("m"_1 + "m"_2)^2 - ("m"_1 - "m"_2)^2 = 4"m"_1"m"_2`

∴ `(4"h"^2)/"b"^2 - ("m"_1 - "m"_2)^2 = "4a"/"b"`

`=> "m"_1 - "m"_2 = (2sqrt("h"^2 - "ab"))/"b"`

Given, `"m"_1/"m"_2 = 5/3`

`=> ("m"_1 + "m"_2)/("m"_1 - "m"_2) = (5 + 3)/(5 - 3)`

`=> (-2"h"//"b")/((2sqrt("h"^2 - "ab"))/"b")` = 4

`=> - "h" = 4sqrt("h"^2 - "ab")`

`=> "h"^2 = 16("h"^2 - "ab")`

`15"h"^2 = 16 "ab"`

`=> "h"^2/"ab" = 16/15`