Advertisement Remove all ads

If one of the lines given by ax2 + 2hxy + by2 = 0 is perpendicular to px + qy = 0, show that ap2 + 2hpq + bq2 = 0. - Mathematics and Statistics

Sum

If one of the lines given by ax2 + 2hxy + by2 = 0 is perpendicular to px + qy = 0, show that ap2 + 2hpq + bq2 = 0.

Advertisement Remove all ads

Solution

To prove: ap2 + 2hpq + bq2 = 0.

Let the slope of the pair of straight lines ax2 + 2hxy + by2 = 0 be m1 and m2 

Then, m1 + m2 = `(-2"h")/"b"` and m1m2 = `"a"/"b"`

Slope of the line px + qy = 0 is `(-"p")/"q"`

But one of the lines of ax2 + 2hxy + by2 = 0 is perpendicular to px + qy = 0 

`=> "m"_1 = "q"/"p"`

Now, m1 + m2 = `(-2"h")/"b"` and m1m2 = `"a"/"b"`

`=> "q"/"p" + "m"_2 = (-2"h")/"b"` and `("q"/"p")"m"_2 = "a"/"b"`

`=> "q"/"p" + "m"_2 = (-2"h")/"b"` and `"m"_2 = "ap"/"bq"`

`=> "q"/"p" + "ap"/"bq" = (-2"h")/"b"`

`=> ("bq"^2 + "ap"^2)/"pq" = - 2"h"`

`=> "bq"^2 + "ap"^2 = - 2"h pq"`

`=> "ap"^2 + 2"hpq" + "bq"^2 = 0`

Concept: Homogeneous Equation of Degree Two
  Is there an error in this question or solution?
Advertisement Remove all ads

APPEARS IN

Advertisement Remove all ads
Advertisement Remove all ads
Share
Notifications

View all notifications


      Forgot password?
View in app×