Elements of analytical geometry and the differential and integral calculus . ce its absolute projection ispractically impossible. 3. Construct the line whose equation is y=\-5. 4. Construct the line whose equation is y=—3x—3. PHOPOSITION II.—PROBLEM. To find the distance between two given points in the plane ofthe co-ordinate axes. Also, to find the angle made by the linejoining the two given points, and the axis of X. Definition.—A point is said to be given when its co-ordinatesare known. Known co-ordinates are designated by x\ y, — xy— xy y ; which are read x prime, x second, &c. When t
Elements of analytical geometry and the differential and integral calculus . ce its absolute projection ispractically impossible. 3. Construct the line whose equation is y=\-5. 4. Construct the line whose equation is y=—3x—3. PHOPOSITION II.—PROBLEM. To find the distance between two given points in the plane ofthe co-ordinate axes. Also, to find the angle made by the linejoining the two given points, and the axis of X. Definition.—A point is said to be given when its co-ordinatesare known. Known co-ordinates are designated by x\ y, — xy— xy y ; which are read x prime, x second, &c. When the point designated by the co-ordinates i^ no particularone, we write simply x and y, to represent its co-ordinates. Let the two given points be Pand Q, and because the point Pis said to be given, we know thetwo distances -4i\^and \ NP=y\ And because the point Q isgiven we know the two distances AM=x and MQ=:y\ AM—AN=:NM=:PR=x—x\ MQ—ME= QE=y—y\ In the right angled triangle PJRQ we have {PPy+(PQy=(PQ)». Put J)=PQ. That is, D=(x—xy+(y—yy,. Or D=^(x—xy+(y-y)K Thus we discover that the distance between any two givenpoints is equal to the square root of the sum of the squares of th$difference of their abscisses and ordinaies. STRAIGHT LINES. 16 If one of these points be the origin or zero point, then «=0or y=0, and we have a result obviously true. To find the angle between PQ and AX. PR is drawn parallel to AX, therefore the angle sought is thesame in value as the angle QPR. Designate the tangent of this angle by a, then by trigonometry we have PR \ RQ \\ radius : tan. QPR. That is, x—x : y—y ::!:«. Whence a=^ -. x—x In case y=^y, PQ will coincide with PR, and be parallel to AX, and the tangent of the angle will then be 0, and this is shown by the equation, for then «=_JL_=o. x x Incase x=.x, then PQ will coincide with RQ and be paral-lel to A Y, and tangent a will be infinite, and this too the equa-tion shows, for if we make x=^x or x—.t
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