Mathematics Exam  >  Mathematics Tests  >  Topic-wise Tests & Solved Examples for Mathematics  >  Test: Dimensional Geometry - 8 - Mathematics MCQ

Test: Dimensional Geometry - 8 - Mathematics MCQ


Test Description

20 Questions MCQ Test Topic-wise Tests & Solved Examples for Mathematics - Test: Dimensional Geometry - 8

Test: Dimensional Geometry - 8 for Mathematics 2024 is part of Topic-wise Tests & Solved Examples for Mathematics preparation. The Test: Dimensional Geometry - 8 questions and answers have been prepared according to the Mathematics exam syllabus.The Test: Dimensional Geometry - 8 MCQs are made for Mathematics 2024 Exam. Find important definitions, questions, notes, meanings, examples, exercises, MCQs and online tests for Test: Dimensional Geometry - 8 below.
Solutions of Test: Dimensional Geometry - 8 questions in English are available as part of our Topic-wise Tests & Solved Examples for Mathematics for Mathematics & Test: Dimensional Geometry - 8 solutions in Hindi for Topic-wise Tests & Solved Examples for Mathematics course. Download more important topics, notes, lectures and mock test series for Mathematics Exam by signing up for free. Attempt Test: Dimensional Geometry - 8 | 20 questions in 60 minutes | Mock test for Mathematics preparation | Free important questions MCQ to study Topic-wise Tests & Solved Examples for Mathematics for Mathematics Exam | Download free PDF with solutions
Test: Dimensional Geometry - 8 - Question 1

x2 + y2 - 2hx - 2ky + h2 = 0 is the equation of the circle whose centre is

Detailed Solution for Test: Dimensional Geometry - 8 - Question 1

The given equation of the circle is
x2 + y2 - 2hx - 2ky + h2 = 0
or (x-h)2 + (y-k)2 - k2= 0
or (x - h)2 + (y - k)2 = k2
Thus the radius = y- coordinate of the centre and therefore circle touch the axis of x.

Test: Dimensional Geometry - 8 - Question 2

 The equation x2 - y2 - 2gx - 2fy + c = 0 represents a circle whose centre is

Detailed Solution for Test: Dimensional Geometry - 8 - Question 2

The given equation of the circle is

1 Crore+ students have signed up on EduRev. Have you? Download the App
Test: Dimensional Geometry - 8 - Question 3

The general equation of second degree ax2 + 2hxy - by2 + 2fy + c = 0 represents a circle if

Detailed Solution for Test: Dimensional Geometry - 8 - Question 3

The general equation

Test: Dimensional Geometry - 8 - Question 4

How many independent geometrical conditions are required for determining a circle?

Detailed Solution for Test: Dimensional Geometry - 8 - Question 4

The general equation of the circle
x2 + y2 + 2gx + 2fy + c = 0
involves three unknowns, namely, g, f and c. Therefore exactly three conditions are required for determining the circle and no more.
Remark: It follows that a unique circle will always pass through three non-collinear points in the plane.

Test: Dimensional Geometry - 8 - Question 5

The equation of the circle whose diameter is the line segment joining (x1, y1) and (x1, y2) is given

Detailed Solution for Test: Dimensional Geometry - 8 - Question 5

Proof: Let the circle be described on the line segment joining A(x1, y1) and B(x2, y2) as diameter. Consider a point P(x, y) on the circle (other than A and B). Then we know from geometry that

This is the required equation.
Comments: Centre C of this circle is the middle point of AB

Test: Dimensional Geometry - 8 - Question 6

The intercept made on the x axis by the circle ax2 - ay2 + 2gx - 2fy + c = 0 is given b

Detailed Solution for Test: Dimensional Geometry - 8 - Question 6

Let the circle cut the axis of x at the points A and R. Then
y = 0 for these points.
Substituting y = 0 in the given equation of the circle, we get

Test: Dimensional Geometry - 8 - Question 7

The equation of the tangent at any point (x' , y') on the circle X2 - y2 + 2gx = 2fy - c = 0. is given by

Detailed Solution for Test: Dimensional Geometry - 8 - Question 7

Remark: Note that in obtaining the equation of the tangent to the circle at a given point (V ,i j ) , we replace in the equation of the circle as follows:

Test: Dimensional Geometry - 8 - Question 8

The line y = mx + c cuts the circle x2 + y2 = a2, in two points only if

Detailed Solution for Test: Dimensional Geometry - 8 - Question 8

We are given the following:

Their points of intersection will be obtained by- solving equations (i) and (ii). Eliminating y from (i) and (ii), we get 

The roots of this equation and hence the points of intersection of (i) and (ii) are real only if 

Test: Dimensional Geometry - 8 - Question 9

The length of the chord intercepted by the circle x2 + y2 = a2 on the straight line y = mx + c is given by

Test: Dimensional Geometry - 8 - Question 10

The line y = mx + c is a tangent to the circle x2 + y2 = a2 at the point on the circle whose coordinates are given by

Test: Dimensional Geometry - 8 - Question 11

The equation of a straight line passing through the point (x1, y1, z1) having direction cosines as l, m, n is given by

Detailed Solution for Test: Dimensional Geometry - 8 - Question 11

Equation of a line passing through a point (x1, y1, z1) and having d.c.’s [l,m,n].
Let P(x, y, z) be any point on the line.
Then the d.c.’s of the line joining A and P are


This is th required equation of the lline.

Test: Dimensional Geometry - 8 - Question 12

If the lines

are perpendicular to each other, then k is equal to 

Detailed Solution for Test: Dimensional Geometry - 8 - Question 12

Proof: the d.c.’s of the two given lines are [3, 2k, 2] and (3k, 1,-5) Since the lines are perpendicular, therefore

Test: Dimensional Geometry - 8 - Question 13

What are the coordinates of the point of intersection of the line   3x+4y+5z=5?

Detailed Solution for Test: Dimensional Geometry - 8 - Question 13


Test: Dimensional Geometry - 8 - Question 14

The straight line passing through (a,b, c) and parallel to the z-axis is given by

Test: Dimensional Geometry - 8 - Question 15

The straight line passing through (a, b, c) and perpendicular to x-axis is given by

Detailed Solution for Test: Dimensional Geometry - 8 - Question 15

The d.c.’s of the line perpendicular to x-axis are (0, m, n)
∴ Required equation will be

Test: Dimensional Geometry - 8 - Question 16

To transform the equations of a line from unsymmeirieal form to the symmetrical form, it is necessary to know about

Detailed Solution for Test: Dimensional Geometry - 8 - Question 16

The equations of the line in the symmetrical form are

where (x1, y1 ,z1) is a given point on the line and (l, m, n) are the direction ratios of the line. Therefore the equations of a straight line can be written in symmetrical form if we know.
(i) the direction ratios of the line and
(ii) a point on it.

Test: Dimensional Geometry - 8 - Question 17

The equations of the line through the point (1,2,4) and parallel to the line 3x + 2y - z = 4, x - 2y - 2z = 5 is given by

Detailed Solution for Test: Dimensional Geometry - 8 - Question 17

 We know that any two equations of first dgree in x, y and z taken together represent a line [two non parallel planes always intersect in a line].
The equations of the line are

i.e.  the line is the intersection of planes (i) and (ii) Let (l, m, n) be the d.c.'s of the line of intersection of planes (i) and (ii). Since the line is common to both the planes, therefore it is perpendicular to the normals to the two planes. But the direction ratios of these normals are the coefficients of x, y and x in the equations of the planes. Therefore we have

Test: Dimensional Geometry - 8 - Question 18

The angle between the lines in which the planes 3x - 7y - 5z = 1 and 5x - 13y + 3x + 2 = 0 cut the plane 8x - 11y + 3z = 0, is equal to

Detailed Solution for Test: Dimensional Geometry - 8 - Question 18

Let (l1, m1, n1) be the d.c.’s of the line of intersection of the planes


Test: Dimensional Geometry - 8 - Question 19

The angle between the line  and the plane ax+by+cz+d=0, is given by

Detailed Solution for Test: Dimensional Geometry - 8 - Question 19

To find the angle between the line


Let θ be 1 he angle between the line (i) and the plane (ii). Then 90 - θ is the angle between the line and the normal to the plane.
Now th2 d.r.’s of the line (i) are l, m, n and the d.r/s of the line perpendicular lo (ii) are a, b, c
∴ cos (90 - θ)

Test: Dimensional Geometry - 8 - Question 20

If the straight line   is parallel to the plane ax + by + cz + d = 0, then

27 docs|150 tests
Information about Test: Dimensional Geometry - 8 Page
In this test you can find the Exam questions for Test: Dimensional Geometry - 8 solved & explained in the simplest way possible. Besides giving Questions and answers for Test: Dimensional Geometry - 8, EduRev gives you an ample number of Online tests for practice
Download as PDF