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

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

Please post the solution in DETAIL...

#Algebra #Geometry #StraightLines

Easy Math Editor

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

So putting y=0 in given equation

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

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

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

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

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

Deepanshu Gupta - 5 years, 2 months ago

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

Parag Zode - 5 years, 2 months ago

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 - 5 years, 2 months ago

Sir but can we assume that for finding value of q the equation $6x^2-24x+q=0$ should be perfect square ? If it is then the value of q is equal to 24. This is a JEE Question .

Parag Zode - 5 years, 2 months ago

Are there 2 lines? I see only 1 equation.

Guiseppi Butel - 5 years, 2 months ago

The equation above is multiplication of those 2 lines

Krishna Sharma - 5 years, 2 months ago

What are the 2 lines?

Guiseppi Butel - 5 years, 2 months ago

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

Krishna Sharma - 5 years, 2 months ago

@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 - 5 years, 2 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$