Test: Coordinate Geometry- 1

Step 1 & 2 – Understand the question and draw inferences from the question statement.

Given: lines a and b are in the graph

To find: Comparison of the slopes of the lines

 Step 3 – Analyze Statement 1 Independently

Per statement 1, line a does not intersect quadrant III

Inference 1: Line a can have a zero slope, a negative slope or an undefined slope. However, it cannot have a positive slope.

Since Statement 1 does not tell you anything about line b , the information provided in Statement 1 is not sufficient to arrive at a unique answer.

Step 4 – Analyze Statement 2 Independently

Per Statement 2, line b intersects quadrant IV

Inference 2: line b has any kind of slope

Inference 3: if the slope is positive, the y-intercept is negative

Since Statement 2 does not tell you anything about line a , the information provided in Statement 2 is not sufficient to arrive at a unique answer.

Step 5 – Analyze Both Statements Together

To analyze both statements together, we will put the inferences from both statements. Here is the data that we have:

Inference 1: Line a can't have a positive slope.

Inference 2: line b has any kind of slope

Inference 3: if the slope is positive, the y-intercept is negative

Though we know the slope of line a is negative, we know nothing of the slope of line b . Inference 3 does not help. The question cannot be answered.

 Answer: Option (E)

Step 1: Question statement and Inferences

A circle tangent to the x-axis at 2 and the y-axis at 2 looks like this:

You know that the radius of the circle is 2, making the area of the circle 4π.

Step 2: Finding required values

To find the region between the circle and the origin, measure the 2 × 2 square, and subtract the quarter circle from it. If the area of the circle is 4π, then the quarter circle is π.

Step 3: Calculating the final answer

4 – π is the area between the circle and the origin.

π  = 3.14 and 4 – 3.14 is less than 1.

Answer: Option (A)

Steps 1 & 2: Understand Question and Draw Inferences

If L₁ passes through points in each quadrant except the III^rd quadrant, then the Line must have both its x and y intercepts as positive numbers.

Let’s assume the x-intercept = a and y-intercept = b for the line L₁, where both a and b are positive numbers. This means that the line L₁ passes through (a, 0) and (0, b).

Slope of  . Since a and b are positive, the slope is a negative number.

Step 3: Analyze Statement 1

Slope of  

m₂ is positive since a and b are positive.

SUFFICIENT.

Step 4: Analyze Statement 2

If the slope of line L₁ is negative and it intersects with a line L₂ in the first quadrant, then the slope of L₂ can be positive or negative as shown in the figures below:

The slope of L₂ in the first diagram is negative while in the second quadrant the slope of L₂ is positive.

INSUFFICIENT.

Step 5: Analyze Both Statements Together (if needed)

Since we get an answer from Step 3, there is no need to combine the statements.

(A) is the answer.

Steps 1 & 2: Understand Question and Draw Inferences

To find the area of the triangle, you need the base and the height.

With point P on the x-axis, knowing points Q and R will help you find the base and height.

Step 3: Analyze Statement 1

(1) Point Q lies on the x-axis 6 units from point P.

This gives you the base of the triangle. Whether point Q is left or right of point P, the triangle base is 6. However, you still need the height.

Statement 1 is not sufficient.

Step 4: Analyze Statement 2

(2) Point R is 5 units away from the x-axis

If the base of the triangle is on the x-axis, then knowing point R is 5 away gives you a height of 5. However, with this statement alone, you don’t have the base, and you don’t know whether the base is on the x-axis..

Statement 2 is not sufficient.

Step 5: Analyze Both Statements Together (if needed)

From statement 1, you know that the base is on the x-axis and that it has a length of 6.

From statement 2, because the base is on the x axis, you know the height is 5.

These are all you need to calculate the area of the triangle. The left-right placement of points Q and R don’t affect the area calculation:

Area = ½ * 6 * 5 = 15

Answer: Option (C)

Steps 1 & 2: Understand Question and Draw Inferences

To find the distance from the origin, you need the absolute values of the (x, y) coordinates of points A and B.

To compare the distances of points A and B, you can use the ratios of the (x, y) values of these points.

Step 3: Analyze Statement 1

(1)       |x| = |y| for point A and |x| = |y| for point B

With statement 1, the |x|, |y| values of points A and B could be the same, or they could be different. 

For instance, A could be (3, -3) and B could be (3, 3) or (4, -4).

Statement (1) is not sufficient.

Step 4: Analyze Statement 2

(2)       |x|, |y| of point A = |x|, |y| of point B

This tells you that points A and B are the same distance from the origin. You don’t know which quadrant the points are in, but you can still answer the question.

Statement (2) is sufficient.

Step 5: Analyze Both Statements Together (if needed)

You get a unique answer in step 4, so this step is not required

Answer: Option (B)

Steps 1 & 2: Understand Question and Draw Inferences

We have to determine the specific quadrant in which the point (x, y) lies.

Step 3: Analyze Statement 1

Every point that lies in the II^nd Quadrant has a negative x-coordinate and a positive y-coordinate. Since (x, y + 1) lies in the II^nd Quadrant

x < 0    and     y + 1 > 0

x < 0    and     y > -1

Is x negative or positive? Negative.

Is y negative or positive? Since y is greater than -1, it can hold both negative and positive values.

If x is negative and y is negative then the point (x,y) lies in the III^rd Quadrant.

If x is negative and y is positive then the point (x, y) lies in the II^nd Quadrant.

INSUFFICIENT.

Step 4: Analyze Statement 2

Every point that lies in the III^rd Quadrant has a negative x-coordinate and a negative y-coordinate. Since (x + 3, y) lies in the III^rd Quadrant

x + 3 < 0  and  y < 0

x < -3       and   y < 0

Is x negative or positive? Since x is smaller than -3, it would always be a negative number.

Is y negative or positive? Negative.

Since x and y are both negative, the point (x, y) must lie in the III^rd Quadrant.

SUFFICIENT.

Step 5: Analyze Both Statements Together (if needed)

Since we have obtained an answer, there is no need to combine the statements.

(B) is the answer.

Step 1: Question statement and Inferences

Drawing the axes and the lines y = k and x = h given in the question on xy-plane:

Based on the figure drawn by us, plotting the point R whose coordinates are (-h, -k):

Step 2:Finding required values

Since the line x = h is the perpendicular to QR, the y-coordinate of point R must also be equal to –k. RA and QA are perpendicular to the line x = h, the y-coordinate of R, A and Q are equal. Therefore, the coordinates of point A are (h, -k).

RA = AQ = h – (-h) = 2h

The coordinates of point Q are (h + 2h, -k) or (3h, -k).

Step 3: Calculating the final answer

Since the line y = k is perpendicular to PQ, the x-coordinate of P, B and Q are equal, i.e. 3h.

Since QB = BP, and QB = k – (-k) = 2k, the y-coordinate of point P is k + 2k = 3k.

The coordinates of the point P are (3h, 3k).

(D) is the correct answer.

Steps 1 & 2: Understand Question and Draw Inferences

To find whether the slope is greater than 2, you need information on the orientation of the line. This can come from two points of the line or at least one point that the line passes above or below.

Step 3: Analyze Statement 1

(1) Point P has an x-value of 1 and is 2 units away from the origin

Statement 1 tells you that from the origin, point P creates a right triangle. Use the Pythagorean Theorem to find the y-value: (The triangle base is the x-value of 1, the hypotenuse is the distance of 2, and the height is the y-value)

The height of the triangle, which is the y-value of point p, has an absolute value of √3, making it either √3 or –√3. However, you need two points to draw the line, and you don’t know whether point P is in quadrant I or IV.

Statement 1 is not sufficient.

Step 4: Analyze Statement 2

(2) The line does not pass through Quadrant IV

Statement 2 tells you that the slope is positive (to avoid Quadrant IV), but you don’t know whether it’s greater than 2.

Statement 2 is not sufficient.

Step 5: Analyze Both Statements Together (if needed)

From statement 1, you know the (x, y) values of point P are either (1, √3) or (1, –√3), placing point P in either Quadrant I or IV.

From statement 2, you know that point P is in Quadrant I. You also know the line cannot cross into Quadrant IV.

As shown above, the slope is√3. The line can rotate clockwise, making the slope less, but it cannot rotate counterclockwise, as that would bring it into Quadrant IV. Taken together, you know that the slope is not greater than 2.

Statements 1 and 2 together are sufficient.

Answer: Option (C)

Steps 1 & 2: Understand Question and Draw Inferences

In order to determine any information about the area of a triangle, we must try to obtain information about its base and height since

the area of a triangle =  (1/2)*base*height

If the base is taken as the length of line segment BC, then the height is the y-coordinate of point A.

Step 3: Analyze Statement 1

If the point A lies above the line y = 5 and the side BC lies on the x-axis, then the height of the triangle is more than 5 units. We have no information about the length of the base BC.

INSUFFICIENT.

Step 4: Analyze Statement 2

Let’s assume the coordinates of point B to be (b, 0) and the coordinates of point C to be (c, 0).

If the mid-point of B and C is (6, 0) then

(b+c)/2 = 6

c = 12 – b

Length of BC = 12 – b – b = 12 – 2b

Since OB < BC, we have

b < 12 – 2b

b < 4

We cannot infer anything about the base BC from this, apart from the fact that the length of BC will always be greater than 4 units.

There is also no information about the height of the triangle ABC.

INSUFFICIENT.

Step 5: Analyze Both Statements Together (if needed)

Combining statements (1) and (2),

We have Height > 5 and b < 4

We have information about the height but still we are unsure about the base BC and so we cannot conclude if the area of the triangle ABC is greater than 12 or less than 12.

INSUFFICIENT.

(E) is the correct answer.

Steps 1 & 2: Understand Question and Draw Inferences

Let’s find the x-intercept of the given line.

When the line intercepts the x-axis, the y-coordinate becomes 0.

Substituting y = 0 in the given equation of the line,

The question thus becomes

Multiplying by -1 on both sides of the inequality, the question modifies to

Step 3: Analyze Statement 1

To compare the given line with the general form of the equation (y= mx + c), we divide both sides of the equation with m².

The equation becomes y

we see that the y-intercept of the given line m²y = bx + m is 1/m

The statement tells us that 1/m>0

Using this information, we can only say that m > 0.

However, b/m can be negative or not depending on whether b is negative or not.

INSUFFICIENT.

Step 4: Analyze Statement 2

In order to find the x-intercept of the given equation of line my =  x + m, we substitute y = 0 in the given equation to get

Since  only when b < 0.

Is b/m < 0 ?  We cannot be sure because we only know that b < 0. b/m

can be less than 0 or not depending on whether m is positive or not.

Step 5: Analyze Both Statements Together (if needed)

Combining statements (1) and (2),

we have m > 0 and b < 0.

This means that b/m < 0 for all such values of m and b.

SUFFICIENT.

(C) is the answer.