CameraIcon
CameraIcon
SearchIcon
MyQuestionIcon
MyQuestionIcon
1
You visited us 1 times! Enjoying our articles? Unlock Full Access!
Question

If θ is the acute angle between the lines represented by equation ax2+2hxy+by2=0, then prove that tanθ=2h2aba+b,a+b0.

Open in App
Solution

Let m1 and m2 be the slopes of the lines by the equation

Given, ax2+2hxy+by2=0 ....... (i)

Let y=m1x and y=m2x

(ym1x)(ym2x)=0

and m1m2x2(m1+m2)xy+y2=0 ......... (ii)

Comparing (i) and (ii), we get

m1m2a=1b=m1+m22h

m1m2=ab and m1+m2=2hb

Thus, (m1m2)2=(m1+m2)24m1m2

(m1m2)2=(2hb)24(ab)

(m1m2)2=4(h2ab)b2

Let the angle between y=m1x and y=m2x be θ.

tanθ=m1m21+m1m2

=∣ ∣m1m221+m1m2∣ ∣

=∣ ∣ ∣ ∣ ∣ ∣4(h2ab)b21+ab∣ ∣ ∣ ∣ ∣ ∣

=∣ ∣2(h2ab)a+b∣ ∣, if a+b0

flag
Suggest Corrections
thumbs-up
0
Join BYJU'S Learning Program
similar_icon
Related Videos
thumbnail
lock
Parametric Representation-Ellipse
MATHEMATICS
Watch in App
Join BYJU'S Learning Program
CrossIcon