Rajkiranmareedu wrote:
Karishma: I didn't understand. Could you explain a bit.
VeritasPrepKarishma wrote:
rockroars: I appreciate the effort for the diagrams. But you made a tiny judgment error. Let me explain the answer in detail.
First of all, notice that ax + by – c = 0 or ax + by = c is the equation of the same line.
A line divides the plane into two regions. One of them, where every point (x, y) satisfies ax + by ≥ c, is region R.Statement 1: Slope of line is 2
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Ques1.jpg
the line will pass through third quadrant and hence both regions will lie in the third quadrant. Sufficient.Statement 2: The line passes through (-3, 0).
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Ques2.jpg
A line passing through (-3, 0) could be the blue line or the green line. In either case, the line will pass through the third quadrant and hence, will have both regions in the third quadrant. So it is sufficient too? What about the x axis? That is also a line passing through (-3, 0). It does not pass through the third quadrant. We would need the equation of the line to find out whether our region R lies in the third quadrant. The equation of x axis is y = 0. So the required region is y ≥ 0 i.e. the first and second quadrant. Hence using just this information, we cannot say whether a point of region R lies in the third quadrant or not.
Answer (A)
A line (which by definition, extends infinitely at both sides) given by ax+by - c = 0 splits a region into two sections - one on the left side of the line and the other on the right side of the side. One of these two regions will satisfy ax+by - c < 0 and the other will satisfy ax+by - c > 0.
How do you know which region satisfies which inequality? Put a point from that region in the inequalities and see what it satisfies e.g. if (0, 0) doesn't lie on the line, put x = 0, y = 0
If c is negative, ax+by - c < 0 will be satisfied and the region that contains the point (0, 0) i.e. the origin of the axis will satisfy ax+by - c < 0.
In that case, the other region will satisfy ax+by - c > 0.
Which quadrants will lie in any particular region depends on where the line is located. If it is a vertical line passing through first and fourth quadrant, second and third quadrant will lie to its left and some part of first and fourth quadrants will also lie to its left. Rest of the first and fourth quadrants will lie to its right (make a line and see what i mean)
Similarly, try making a line with a positive slope, say 2. It will pass through first and third quadrants in ALL cases (remember, a line extends indefinitely at both ends). Since it will pass through the third quadrant, both the regions will have a part of the third quadrant.
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