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 Slope-intercept 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? Re-arrange x+ y 6 = 0into slope-intercept 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 slope-intercept 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 Re-formulate 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, // x-y= 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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