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Question

The equation ax3−9yx2−y2x+4y3=0 represents three straight lines. If two of the lines are perpendicular to each other, then the value of a is:

A
5
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B
-5
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C
4
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D
-4
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Solution

The correct options are
A 5
D -4
The given equation is ax39yx2y2x+4y3=0, which represents three straight lines.

We can see that all the lines passes from origin (0,0)

Let's assume the lines are given by equation y=mx

Putting y=mx in the given equation of three straight lines, we get,

ax39(mx)(x2)(mx)2x+4(mx)3=0

(a9mm2+4m3)x3=0

4m3m29m+a=0 ....(1)

This equation in m has three roots, m1, m2 and m3, which are three slopes of three lines respectively.

If the two lines are perpendicular then let's assume m1.m2=1,

Product of roots in equation (1) is m1.m2.m3=a4

Hence m3=a4

also from equation (1), m1+m2+m3=14

m1m2+m3(m1+m2)=94

m1+m2=1a4

a(1a)=20

By Solving the above equation we get a=5,4

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