Elements of analytical geometry and the differential and integral calculus . EOMETRY. infinite number of straight lines can be drawn througb thepoint P. We may give to y and x their numerical values, and give anyvalue whatever to a, then we can construct a particular line thatwill run through the given point P. Suppose a;=2, ?/==3, and make a= the equation will become y—3=4(ar—2).Or 3/=4a?—5. This equation is obviously that of a straight line, hence (3)IS of the required form. PROPOSITION lY. To find the equation of a line which passes through two givenpoints. More definitely, we say fin
Elements of analytical geometry and the differential and integral calculus . EOMETRY. infinite number of straight lines can be drawn througb thepoint P. We may give to y and x their numerical values, and give anyvalue whatever to a, then we can construct a particular line thatwill run through the given point P. Suppose a;=2, ?/==3, and make a= the equation will become y—3=4(ar—2).Or 3/=4a?—5. This equation is obviously that of a straight line, hence (3)IS of the required form. PROPOSITION lY. To find the equation of a line which passes through two givenpoints. More definitely, we say find theequation of the line which passesthrough the two given points Pand Q. As the equation is to be thatof a line, it must correspond toy=ax-\-h. (1)As it must pass through thegiven point P, whose co-ordinatesare x and y, we must havey^=ax-\-h. (2)Subtracting (2) from (1) we have y—y=.a{x—x). (3) Because the line must also pass through the other point Q, wemust have (Prop. II.) a=^—^^. x X Substituting this value of a in (3) we have \x X / the equation STRAIGHT LINES. 17 PROPOSITION V. To find the equation of a straight line which shall pass througha given point and make a given angle with a given line. The equation of the given line must be in the form y^ax-\-h. (1) Because the other line must pass through a given point itsequation must be (Prop. III.) y—y^a{x—x). (2) We have now to determine the value of a. When a and a are equal, the two lines must be parallel, andthe inclination of the two lines will be greater or less accordingto the relative values of a and a. Let PQhQ the given line (thetangent of its angle with the axisof X equal a) and PR the otherline which shall pass throughthe given point P and make agiven angle QPR. The tangentof the angle PRX=:a. Because PRX=PQR-{-QPR. QPR=^PRX—PQR. Tan. QPR=tan.(PRX—PQR.) As the angle QPR is supposed to be known or given, we mayput m to designate its tangent, and m is a known by trigonome
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