find the equation of pair of lines intersecting at (2,-1) and parallel to the pair 6x^2-13xy-5y^2
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MATHS
Show that the following equation represents a pair of straight lines. Find the points of intersection and the acute angle between them.
(i) 3x
2
+7xy+2y
2
+5x+5y+2=0
(ii) 2x
2
−13xy−7y
2
+x+23y−6=0.
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ANSWER
for a general eq ax
2
+2hxy+by
2
+2gx+2fy+c=0 will represents pair of straight line if:
a,b,h
=0 and h
2
−ab>or=0
for 3x
2
+7xy+2y
2
+5x+5y+2=0 a=3,h=7/2,b=2
h
2
−ab=49/4−6=25/4>0 (represents straight line)
for 2x
2
−13xy−7y
2
+x+23y−6=0
a=2,h=−13/2,b=−7
h
2
−ab=169/4+14=225/4>0 (represents straight line)
Points of intersection is α=
ab−b
2
hf−bg
and β=
ab−h
2
hg−af
after substituting values of a,b,h,g,f we will get the points of intersection for the given lines.
acute angle θ=cos
−1
(a−b)
2
+4h
2
∣a+b∣
again, after putting the values of a,b,h will get the acute angle.
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