Tangent line of a circle can be determined once the tangent point or the slope of the line is known.


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1 Worksheet 7: Tangent Line of a Circle Name: Date: Tangent line of a circle can be determined once the tangent point or the slope of the line is known. Straight line: an overview General form : Ax + By + C = 0 Slopeintercept form: y = mx+ c, where slope ( x, y) are points on the line; c is the intercept. y y Δy = = Δ, ( x, y) and m x x x Example : Determine the equation of a line that passes through P(,5) and (4,)! General form: P(,5) Ax + By + C = 0 A+ 5B + C = 0 (4, ) Ax + By + C = 0 4A B + C = 0 Eliminate C: A+ 5B+ C = 0 4A B + C = 0 6A + 6B = 0 A= B Substitute: A+ 5B + C = 0 A+ 5A+ C = 0 A + C = 0 C = A Let: A = B =, C = So, the line is: x + y = 0 Slope  intercept form Δy 5 ( ) 6 m = = = = Δx 4 6 Slope =  Substitute m and one point : P(,5) y = mx + c 5 = ( )( ) + c c = Intercept = Hence, the line is: y = x + Is it the same result as in the general form? Example : What is the slope of x + y 6 = 0? Rearrange x+ y 6 = 0into slopeintercept form: 6 x x+ y 6 = 0 y = y = x y = x+ { { c Hence, the slope is. m Page of 9
2 Exercise. Determine the equation of a line that passes through the following points! a. P(,) and (,) f. A(0,) and B(,) b. M(,) and N(,) g. B(,4) and C(,4) c. (,5) and R(,7) h. C(,) and D(,) d. S (,) and T(,) i. D(,) and E(,) e. U(0,0) and V(0,0) j. E(,5) and F(5,). Determine the slope and intercept of lines in question! Slope Intercept a b c d e f g h i Page of 9
3 j. Determine the lines with slope m and passes through the point Z. m =, Z = (, 5) Does this line pass through (0,)? m =, Z = (0, ) Does this line pass through (,)? m =, Z = (,6 ) Does this line pass through (,9)? m =, Z = (,5 ) 5 Does this line pass through (,5)? 5 m =, Z = (, 6) 6 Does this line pass through (,9)? 4. Rewrite the following line equations into slopeintercept form! a. x+ 7y 5 = 0 b. 0 x+ y = c. x 7y + 5 = 0 d. x y = 0 5. Rewrite the following line equations into general form! b. y + 5 = 4 a. x = y+ 0 5 Page of 9
4 c. y x = 4 d. 5x = 0y Lines and Slopes Let: Line is g with slope of m Line is g with slope of m Then: g // g, i.e. g and g are parallel if and only if m = m g g, i.e. g and g are perpendicular if and only if m m = Example: Find the equation of a line that is parallel to x + y = and passes through (,0) x Line : x+ y = y = y = x+ slope = m =. Line : parallel to line m = m = m = Equation of line : slope and passes through (,0) is 0 ( ) y mx c = + = + c c = y = x or y + x + = 0 Example : Find the equation of a line that is perpendicular to x + y = and passes through (,0). Line : x + y = slope = m = Line : perpendicular to line mm = m = m = Hence; line is y = mx + c 0 = + c c = 7 y = x + 7 Example : Prove that x+ y = and y x+ 00 = 0are perpendicular! Proof: Line : x+ y = y = x+ m = Page 4 of 9
5 x 00 Line : y x+ 00 = 0 y = y = x 50 m = Since mm = = x+ y = y x+ 00 = 0 EXERCISE Find the equation of lines that:. Parallel to y + x = 5and. Perpendicular to y + x = 5and passes through (,) passes through (,). Parallel to x y + 7 = 0and 4. Perpendicular to x y + 7 = 0and passes through (0,) passes through (0,) 5. Parallel to x = + yand passes through (,) 6. Perpendicular to x = + yand passes through (,) Examine whether these lines are parallel or perpendicular!. x y + 7 = 0 and 4y x + = 0. y = x and y 9x+ = 0 9. x + 5y + 7 = 0 and 5x = y 4. x + y 4 = 0 and y x + = 0 5. x y + 7 = 0 and y x 4 = 0 6. x 5y + 9 = 0 and 4y x + = 0 Page 5 of 9
6 Equation of a Circle s Tangent Line Circle centered at (0,0) and radius r Let: ( x, y ) is ON the circle x + y = r Then: Tangent line of the circle that passes through ( x, y) is given by: x x + y y = r Circle centered at P(a,b) and radius r Let: ( x, y ) is ON the circle ( x a) + ( y b) = r Then: Tangent line of the circle that passes through ( x, y) is given by: x a x a + y b y b = r ( )( ) ( )( ) Example : Find the equation of line that is tangent to the circle x + y = 5 at (,4). The tangent line is xx yy r x y + = 4 = 5 Example : Find the equation of line that is tangent to the circle ( + ) + ( y + ) = 0 x at (,4). The tangent line is x a x a + y b y b = r + x y+ = 0 ( )( ) ( )( ) ( )( ) ( )( ) ( ) ( ) 4 x+ y+ = 0 4x y = 0 x y = 0 Example : Find the equation of line that is tangent to the circle, x + y + x+ y = 0at 4 8 Reformulate the circle into standard form: x + y + 4x+ 8y = 0 x + 4x+ y + 8y = 0 x + x+ y + 4y = 0 ( x + x+ ) + ( y + 4y+ 4) 4 = 0 ( x+ ) + ( y+ ) = Page 6 of 9
7 The tangent line is ( x a)( x a) ( y b)( y b) r ( x ) ( y ) + = = 5 ( x+ ) + ( y+ ) = x+ 5y+ = x+ 5y = EXERCISE. Determine the tangent line of a circle centered at (0,0) with radius r and passes through point M a. r =, M(,) b. r =, M(, ) c. r =, M(, ) d. r =, M(, ). Determine the tangent line of a circle centered at P(a,b) with radius r and passes through point N a. P(0,), r =, N(, ) b. P(, ), r =, N(, ) c. P(,), r =, N(, ) d. P(,), r =, N(,). Determine the tangent line of the following circle and passes through point a) x + y = 00, (6,8) b) 6x + 6y = 5,, 4 c) x + y 6x =0,(0,) d) x + y 4x+ 8y 7= 0, (,) Page 7 of 9
8 Tangent Line of a Circle when the Slope is Known Circle centered at (0,0) and radius r Let Slope of circle x + y = r is m Then The tangent line is given by y = mx± r + m Circle centered at P(a,b) and radius r Let Slope of circle ( x a) + ( y b) = r is m Then The tangent line is given by ( ) y b = m x a ± r + m Example : Find the line with slope 4 that is tangent to the circle x + y = 5 The tangent line is: y = 4x± 5 7 y = 4x+ 5 7 and y = 4x 5 7 Example : Find the tangent line of the circle ( ) ( ) Line y = x + 5 m = x + y+ = 6that is parallel to y = x + 5 So, the tangent line is: y + = x ± 4 5 y = x and y = x EXERCISE ( ). Find the equation of the tangent line of a circle centered at (0,0) and radius r, with slope m. a. r =, m = b. r =, m = c. r =, m = d. r =, m = 4 Page 8 of 9
9 . Find the equation of the tangent line of a circle centered at P(a,b) and radius r, with slope m. a. P(0,), r =, m = b. P(,), r =, m = c. P(,), r =, m = d. P(,), r =, m = 4 a.. Find the equation of the tangent line of the following circles. 6x + 6y = 5, // x + y = 0 b. 6x + 6y = 5, x + y = 0 c. 6x + 6y = 5, // xy= d. + = 6x 6y 5, x y = 4. Find points on the circle ( ) ( 5) tangent line with slope 5. x+ + y = 74 that are contained in the Page 9 of 9
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