Co-ordinate geometry

[sourabh33]

In the xy-coordinate plane, line l1 and line l2 intersect at the point (-2 , -5). Is the product of their slopes negative?

(1) The product of the x-intercepts of lines l1 and l2 is negative.
(2) The product of the y-intercepts of lines l1 and l2 is positive.

[Anurag@Gurome]

In the xy-coordinate plane, line l1 and line l2 intersect at the point (-2 , -5). Is the product of their slopes negative?

(1) The product of the x-intercepts of lines l1 and l2 is negative.
(2) The product of the y-intercepts of lines l1 and l2 is positive.

Solution:
Let the equation of line l1 be y = m1x + c1.
Let the equation of line l2 be y = m2x + c2.
We need to know whether m1*m2 is negative or not.
Let us first consider (1) alone.
x-intercept of l1 is -c1/m1.
x-intercept of l2 is -c2/m2.
So, (c1*c2)/(m1*m2) < 0.
But this does not tell us whether m1*m2 < 0 or not.
Or (1) alone is not sufficient.
Next, consider (2) alone.
y-intercept of l1 is c1.
y-intercept of l2 is c2.
So, c1*c2 > 0.
Again this does not tell us whether m1*m2 < 0 or not.
Or, (2) alone is not sufficient.
Next, combine both the statements together and check.
On combining we have that (c1*c2)/(m1*m2) < 0 and c1*c2 > 0.
This automatically means that m1*m2 < 0.
So, both statements together are sufficient to answer the question.

The correct answer is (C).

[sourabh33]

Cool ! I find the algebraic approach short, simple and time-saving.

My approach (below) was error prone and time consuming.