Can someone tell me how to approach and solve this math problem?

If the pair of lines 6x2pxy3y224x+3y+q=06x^2-pxy-3y^2-24x+3y+q=0 intersect on x- axis then p is equal to :- a)3/2a)3/2 b)5/2b)-5/2 c)18c)-18 d)7d)-7

Please post the solution in DETAIL...

Note by Parag Zode
4 years, 8 months ago

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For to intersect this pair of straight line on x-axis , Let They intersect at point P (α0\alpha \, 0 )

So putting y=0 in given equation

6α224α+q=06\alpha ^{ 2 }\quad -\quad 24\alpha \quad +\quad q\quad =\quad 0.

Now since Intersection point is unique so value of α\alpha is also unique. So Discriminant of this quadratic equation is zero !!

q=24α=2q\quad =\quad 24\\ \\ \alpha \quad =\quad 2.

Therefore Given equation representing pair of straight lines if it's Discriminant is zero !!

6p/212p/233/212324=0P=3/2\left| \begin{matrix} 6 & -p/2 & -12 \\ -p/2 & -3 & 3/2 \\ -12 & 3 & 24 \end{matrix} \right| \quad =\quad 0\\ \\ P\quad =\quad 3/2.

Deepanshu Gupta - 4 years, 8 months ago

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Ok! I didn't knew to use determinant after finding q and a(alpha).. Anyways ,Nice solution.

Parag Zode - 4 years, 8 months ago

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The line(s) intersect the x-axis when y=0. The equation then becomes 6x^2 - 24x + q = 0. There are tons of values for x that will solve this equation depending on the value of q.

If y = 0 then values for p are limitless (infinite).

Guiseppi Butel - 4 years, 8 months ago

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Sir but can we assume that for finding value of q the equation6x224x+q=06x^2-24x+q=0 should be perfect square ? If it is then the value of q is equal to 24. This is a JEE Question .

Parag Zode - 4 years, 8 months ago

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Are there 2 lines? I see only 1 equation.

Guiseppi Butel - 4 years, 8 months ago

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The equation above is multiplication of those 2 lines

Krishna Sharma - 4 years, 8 months ago

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What are the 2 lines?

Guiseppi Butel - 4 years, 8 months ago

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@Guiseppi Butel That is what we have to find in terms of 'p'

Krishna Sharma - 4 years, 8 months ago

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@Krishna Sharma You also have a q which affects the situation.

If this line intersects the x axis then the value for y = 0.

Guiseppi Butel - 4 years, 8 months ago

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