Elements of analytical geometry and the differential and integral calculus . will become nearer a right angle as F approaches Aor A. PROPOSITION VX. To find the equation of a straight line which shall be tangerUto an ellipse. Let X, y, be the co-ordinates ofany indefinite point R, in a linecutting an ellipse ; x, y, the co-ordinates of the point F, and x,y, the co-ordinates of the pointQ. Also, let a be the tangent of the angle of inclination of theline FR with the axis of X. The object is to find the value ofa when FR is tangent to the ellipse. The equation of a line which passes through two
Elements of analytical geometry and the differential and integral calculus . will become nearer a right angle as F approaches Aor A. PROPOSITION VX. To find the equation of a straight line which shall be tangerUto an ellipse. Let X, y, be the co-ordinates ofany indefinite point R, in a linecutting an ellipse ; x, y, the co-ordinates of the point F, and x,y, the co-ordinates of the pointQ. Also, let a be the tangent of the angle of inclination of theline FR with the axis of X. The object is to find the value ofa when FR is tangent to the ellipse. The equation of a line which passes through two points, asR and F, must be of the form y—y=a{x—x). (1) (Prop. Ill, Chap. I.) The equation for the same line passing through the two pointsR and Q, must be • y—y=a{x~x). (2) And the equation for the same line passing through the twopoints F and Q, must be y—y=a{x—xY (3) *In trigonometry we learn that tan. z cot. j:=i22=l. That is, the pro-duct of two tangents the sura of whose arc is 90°, equals 1. When theBum is less than 90°, the product will be a 50 ANALYTICAL GEOMETRY. Because the points P and Q are in the curve, the co-ordinatesof those points must correspond to the following equations : By subtraction A^7/^—f^)-{B^{x^—x^)=0. Or A^y+f )(y-f )=--£(x+x)ix^x). (4) Dividing (4) by (3) we have AHy+f)=---(x+x), (5) ISow conceive the line to revolve on the point F until Q co-inoides with F, then FF will be tangent to the curve. Butwhen Q coincides with F, we shall havey=y and x=x. Whence (5) becomes 2Ay=-^—x\a Or B^x A^y This value of a put in (1) givesB^x y—y {x—x). A^y Reducing Ayy-\-B^xx=^Ay^-\-Bx^. Or A^yy+B^xx=A^BK This is the equation sought, x and y being the general co-ordinates of the line. Scholium 1. To find where the tangent meets the axis of Xwe must make y=0. This gives x=:^^-=CT. X In case the ellipse becomes a cir-cle, £—A, and then the equationwill become yy-\-xx=A^,the equation for a tangent line to a
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