Elements of analytical geometry and the differential and integral calculus . when QPJIbecomes a tangent at P, x=x and y=y, then (7) reduces to i a=-^. (8) THE CIRCLE. 33 This value of a substituted in (5) gives y-y=-<(^-^). (9) y This is the general equation of a tangent line ; x\ y\ arethe co-ordinates of the tangent point, and x, y, the co-ordinatesof any other point in the line. Scholium. For the point in whichthe tangent line cuts the axis of X, wemake y=0, then x=:^=:AT. x For the point in which it meets theaxis of Y, we make x=0, and y=~^=AQ. y Definitions.—A line is said to be normal
Elements of analytical geometry and the differential and integral calculus . when QPJIbecomes a tangent at P, x=x and y=y, then (7) reduces to i a=-^. (8) THE CIRCLE. 33 This value of a substituted in (5) gives y-y=-<(^-^). (9) y This is the general equation of a tangent line ; x\ y\ arethe co-ordinates of the tangent point, and x, y, the co-ordinatesof any other point in the line. Scholium. For the point in whichthe tangent line cuts the axis of X, wemake y=0, then x=:^=:AT. x For the point in which it meets theaxis of Y, we make x=0, and y=~^=AQ. y Definitions.—A line is said to be normal to a curve when itis perpendicular to the tangent line at the point of contact. Join APy and if APT is a right angle, then AP is a normal,and AB, a portion of the axis of X under it, is called the sub-normal. The line BT under the tangent is called the suhtangerU. Let us now discover whether APT\^ or is not a right (8) shows us the tangent value of the inclinationof the line PT with the axis of X. Put a= the tangent of the angle PAT, then by trigonometry. «=^. x But x Eq. (8) Whence aa=L—1. Or a=—i, Therefore AP is at right angles to PT, (Prop. V. Chap. I.) ii ANALYTICAL GEOMETRY. PROPOSITION IV. To find the equation of a line which shall pass through a givenpoint without the circle. Let H be the given point, and x and y its given co-ordinates,and X and y the co-ordinates of the tangent point P. The equation of the line passing through the two points Hand P, must be of the form y—y=a{x—x). (1) And if PH is tangent at the point P, and x and y the co-ordinates of the point P, equation (8) of the last propositiongives us x — -• This value of a put in (1) and we have y—y =—- {x—x)y for the equation sought. This equation combined with that of the circlex^+y^^R^will determine the values of x and y, and as there will be twovalues to each, numerically equal, it shows that two equal tan-gents can be drawn from H, or from any point without the circle,
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