Coordinate Geometry [Part: 3]

If (4, -5) and (-7, 6) are the extremities of a diagonal of a parallelogram and (-1, 2) is its third vertex, then the fourth vertex is:

• (0, 3)

• (-2, -1)

• (2, 1)

• None of these

Answer explanation

To find the fourth vertex of the parallelogram, use the midpoint formula. The midpoint of the diagonal formed by (4, -5) and (-7, 6) is (-1.5, 0.5). Using (-1, 2) as one vertex, the fourth vertex is calculated as (-2, -1).

If the midpoint of the line segment joining the points (-8, 15) and (K, 5) is (a, b), where 3a + 4b = 7, then the value of K is:

• 12

• -10

• 8

• -12

Answer explanation

To find K, use the midpoint formula: M = ((-8 + K)/2, (15 + 5)/2) = (a, b). This gives M = ((-8 + K)/2, 10). Setting 3a + 4b = 7 leads to 3((-8 + K)/2) + 40 = 7. Solving gives K = 12.

The area of a triangle is 10. Two of its vertices are (4, 2) and (6, -1). If the third vertex lies on the line y = x + 4, then the third vertex is:

• (9/2, 15/2)

• (7/2, 11/2)

• (8/2, 14/2)

• None of these

Answer explanation

To find the third vertex, we use the area formula for a triangle. The area is 10, and the vertices are (4, 2) and (6, -1). The third vertex on the line y = x + 4 is (7/2, 11/2), which satisfies the area condition.

Given the points A(P, 5) and B(0, -4), the equation of the locus of the point P(x, y) such that AP - BP = 7 is:

• x^2 + y^2 - 7 = 0

• 2x^2 + 5y^2 = 7

• x^2 + y^2 + 2xy = 9

• None of these

Answer explanation

To find the locus of point P such that AP - BP = 7, we can derive the equation from the distance formula. After simplification, we find that the correct equation is x^2 + y^2 - 7 = 0.

The locus of a point such that the sum of its distances from the points (0, 3) and (0, -3) is 8 is:

• x^2 + y^2 = 25

• x^2 + y^2 = 36

• x^2 + y^2 = 16

• None of these

Answer explanation

The sum of distances from (0, 3) and (0, -3) being 8 describes an ellipse. The foci are at (0, 3) and (0, -3) with a major axis length of 8, leading to a semi-major axis of 4. Thus, the equation is x^2 + y^2 = 36.

The equation of a line with slope -5 and cutting off an intercept of 8 units on the negative direction of the y-axis is:

• 5x + y = 8

• x + 5y = 8

• y = -5x + 8

• None of these

Answer explanation

The line's slope is -5, and the y-intercept is -8 (positive 8 on the negative y-axis). The slope-intercept form is y = -5x - 8. Rearranging gives 5x + y = 8, which matches the correct choice.

Find the equation of the line which cuts off an intercept of 6 on the positive direction of the x-axis and an intercept of 4 on the negative direction of the y-axis:

• x + 3/2y = 1

• 2/3x + y = 1

• x/6 - y/4 = 1

• None of these

Answer explanation

The line intercepts the x-axis at (6,0) and the y-axis at (0,-4). Using the intercept form, the equation is x/6 - y/4 = 1, which matches the correct choice.

If the length of the perpendicular from the origin to a line is 5, and the line makes an angle of 45 with the positive direction of the y-axis, the equation of the line is:

• x + y = 5

• x - y = 5

• y - x = 5

• None of these

Answer explanation

The line makes a 45-degree angle with the y-axis, meaning its slope is -1. The equation in slope-intercept form is y = -x + b. Given the perpendicular distance from the origin is 5, we find b = 5, leading to x - y = 5.

The equation of the straight line that has a y-intercept of 5 and is parallel to the straight line 3x - 2y = 7 is:

• 3x - 2y = 10

• 3x - 2y = 5

• 3x - 2y = -10

• 3x + 2y = 10

Answer explanation

The line 3x - 2y = 7 has a slope of 3/2. A parallel line also has this slope. With a y-intercept of 5, the equation becomes y = (3/2)x + 5, or rearranging gives 3x - 2y = 10, which is the correct choice.

Two opposite vertices of a rectangle are (2, 6) and (7, 1). If the other two vertices of the rectangle lie on the line y - x = 1, then lambda is:

• -2

• 1

• 3

• None of these

Answer explanation

To find the other two vertices of the rectangle, we use the midpoint formula and the line equation y - x = 1. Solving gives us the coordinates of the vertices, leading to lambda = 3 as the correct answer.