Question

Let L be the line joining the origin to the point of intersection of the lines represented by 2x^2 - 3xy - 2y^2 + 10x + 5y = 0. If L is perpendicular to the line kx + y + 3 = 0, then k is equal to ?

Answer 1

Answer :-

We have,

      2x² - 3xy - 2y² + 10x + 5y = 0

⟹ (2x + y)(x - 2y + 5) = 0

Now, equation of the line passing through origin is 2x + y = 0.

Slope of the line is -2.

Since, this line is perpendicular to the line kx + y + 3 = 0.

∴ (-2)(-k) = -1 ⟹ k = -1/2

Answer 2

$\large\underline{\sf{Solution-}}$

Given pair of line is

$\rm :\longmapsto\: {2x}^{2} - 3xy - {2y}^{2} + 10x + 5y = 0$

$\rm :\longmapsto\: ({2x}^{2} - 3xy - {2y}^{2}) + 10x + 5y = 0$

$\rm :\longmapsto\: ({2x}^{2} - 4xy + xy - {2y}^{2}) + 10x + 5y = 0$

$\rm :\longmapsto\:2x(x - 2y) + y(x - 2y) + 5(2x + y) = 0$

$\rm :\longmapsto\:(x - 2y)(2x + y) + 5(2x + y) = 0$

$\rm :\longmapsto\:(x - 2y + 5)(2x + y) = 0$

So two lines are

$\rm :\longmapsto\:x - 2y + 5 = 0 - - - (1)$

and

$\rm :\longmapsto\:2x + y = 0$

$\rm :\implies\:y = - 2x$

On substituting y = - 2x, in equation (1), we get

$\rm :\longmapsto\:x - 2( - 2x) + 5 = 0$

$\rm :\longmapsto\:x + 4x + 5 = 0$

$\rm :\longmapsto\:5x + 5 = 0$

$\rm :\longmapsto\:5x = - 5$

$\bf\implies \:x = - 1$

$\bf\implies \:y = 2$

Let this point of intersection of two lines is represented by coordinate P ( -1, 2 ).

Now, Slope of line L joining the points O and P is given by

$\rm :\longmapsto\:Slope, \: m_1 = \dfrac{2 - 0}{ - 1 - 0}$

$\rm :\longmapsto\:Slope, \: m_1 = \dfrac{2}{ - 1}$

$\rm :\longmapsto\:Slope, \: m_1 = - 2$

Now,

Slope of line kx + y + 3 = 0 is given by

$\rm :\longmapsto\:m_2 = - \dfrac{coefficient \: of \: x}{coefficient \: of \: y}$

$\rm :\longmapsto\:m_2 = - \dfrac{k}{1}$

$\rm :\longmapsto\:m_2 = - \: k$

As given that, line L is perpendicular to kx + y + 3 = 0

$\bf\implies \:m_1 \times m_2 = - 1$

$\bf\implies \: - k \times ( - 2) = - 1$

$\red{\bf\implies \: \boxed{ \sf{ \:k = - \dfrac{1}{2}}}}$

Additional Information :-

1. Slope of line which makes an angle p with positive direction of x axis is given by m = tanp.

2. If line is parallel to x - axis, its slope is 0.

3. If line is parallel to y - axis, its slope is not defined

4. Two lines having slope M and m are parallel, iff M = m

5. If two lines having slope M and m are perpendicular, iff Mm = - 1.