How can one find the values of k for which the line 2x -k is tangent to the circle with the equation x^2 + y^2 = 5?
The simple way to do this is to clearly define what it means for tangent so that finding the k values is the easiest. Problem Specific AnswerWe have [math]y = 2x - k[/math] and [math]x^2 + y^2 = 5[/math] and the line is tangent to the circle. What it means for a 2 things to be tangent is that one point and only one point satisfies both equations at the same time. That means that in the above two equations, only the point represented by [math](x,y)[/math] satisfies both equations. We get: [math]y = 2x - k[/math] and [math]x^2 + y^2 = 5 [/math]Substituting, we get [math]x^2 + (2x-k)^2 = 5.[/math] This equation must only have one solution. [math]5x^2 -4kx +(k^2 - 5) = 0.[/math]This quadratic equation must have its determinant equal to 0 in order for the two to be tangent. [math]b^2 - 4ac = 0[/math][math]16k^2 - 20k^2+100 = 0[/math][math]4k^2 = 100[/math][math]\boxed{k = \pm 5}[/math]Generic AnswerWe can generalize this solution for any line and circle.Line: [math]y = mx+b[/math]Circle: [math]x^2 + y^2 = r^2[/math]Substituting we get: [math]x^2 + (mx+b)^2 = r^2[/math][math]\Rightarrow (m^2 + 1)x^2 + 2mbx + (b^2 - r^2) = 0[/math]Taking the discriminant and making it equal to 0, we get[math]4m^2b^2 - 4(m^2 + 1)(b^2 - r^2) = 0[/math][math]4m^2b^2 - 4m^2b^2 + 4m^2r^2 - 4b^2 + 4r^2 = 0[/math][math]m^2r^2 + r^2 = b^2[/math][math]m, r,[/math] and [math]b[/math] must satisfy these equations in order to be tangent. We can check out previous answer by plugging in here: [math]m = 2; r = \sqrt{5}; b = -k.[/math][math]20 + 5 = k^2.[/math]Therefore, [math]\boxed{k = \pm 5}[/math]
ALGEBRA FACTORING POLYNOMIALS!!!PLEASE HELP!!!?
♣ QUESTION 1 : u^2-6u-72 ♣ SOLUTION : We have to break -6u in such a way that the fragmented terms' product gives us -72u^2,which is the product of the first and third terms. Now,factors of 72 = 1,72,2,36,3,24,4,18,6,12,8,9. We can use 6 and 12. Now, u^2 - 6u - 72 = u^2 plus 6u - 12u - 72 = u (u plus 6) - 12 (u plus 6) = (u plus 6) (u - 12) ♣ QUESTION 2 : 6x^2 plus 7x-3 ♣ SOLUTION : 6x^2 plus 7x - 3 = 6x^2 plus 9x - 2x - 3 = 3x (2x plus 3) -1 (2x plus 3) = (2x plus 3) (3x - 1) ♣ QUESTION 3 : 15y^2 plus 19y-10 ♣ SOLUTION : 15y^2 plus 19y - 10 = 15y^2 plus 25y - 6y - 10 = 5y (3y plus 5) -2 (3y plus 5) = (3y plus 5) (5y - 2) ♣ QUESTION 4 : 8x^2-10x plus 12xy-15y ♣ SOLUTION : 8x^2 - 10x plus 12xy - 15 y = 2x (4x - 5) plus 3y (4x - 5) = (4x - 5) (2x plus 3y) ♣ QUESTION 5 : (2x plus 5)(x-7)=0 ♣ SOLUTION : Either of them must be equal to zero. When,2x plus 5 = 0, 2x = -5 (subtracting 5 from both sides) or, x = -5/2 And when, x - 7 = 0 or, x = 7. So, x = -5/2 or 7. ♣ QUESTION 6 : 2y^2 plus 3y-2=0 ♣ SOLUTION : 2y^2 plus 3y - 2 = 0 or,2y^2 plus 4y - y - 2 = 0 or,2y ( y plus 2) -1(y plus 2) = 0 or, (2y - 1) (y plus 2) = 0 When,2y - 1 = 0 2y = 1 or, y = 1/2 and when,y plus 2 = 0 y = -2 so, y = 1/2 or -2.
Help with algebra problem?
1. The domain only consists of all working x value for the expression. Since x that gives zero denominator makes the whole expression not real, reject that, so the domain is {x | R, x ≠ -½, 7} 2. Switch the top and bottom of the second fraction and multiply altogether. The, simplify to get... 2w^4/5 3. Simplify the radicand's perfect square factors to get... √(36 * 5 * x^6 * (y^14) * y) = 6x³y^7√(5y) 4. Can't be factored.. 5. Combine like terms to get 5x² + 2x - 35 6. -8/(t - 6) 7. Assume that the expression is 2y/7 + 5y/3.... Combine 'em by LCD method to get.. (6y + 35y)/21 = 41y/21 You solve the rest by yourself! I hope this helps!
If p and q are the zeroes of the quadratic polynomial x2+mx+n2, what is the value of p2+pq+q2?
Sum of zeroes/roots p+q=-m____(1)Product of zeroes p*q=n^2____(2)Squaring equation (1) on both sides,p^2 + q^2 + 2pq = m^2_____(3)Substituting value of p*q from equation (2) in (3)p^2 + q^2 + 2n^2 =m^2Therefore, p^2 + q^2 =m^2 - 2n^2So value of p^2 + q^2 +pq=m^2 - 2n^2 +n^2 (since pq=n^2 from (2) )=m^2 - n^2 or=(m+n)(m-n)
Algebra help please! Answer any that you know?
2. Find the quotient. (12x^2 - 10x -12) ÷ (3x + 2) 3. What is the y-intercept for the graph of this line? 3x - 5y = 6 4. Solve for x 6/x + 2/5 = 8 5. Add and simplify. x + 2x/x+1 6. Solve the system of equations. 2x + 3y = 6 x + y = 4 7. Completely factor the polynomial. 12x^2 + 2x - 4 8. Solve for x. 3x - 5 < 7 9. If f = {(2, 3), (5, 7), (3, 3), (5, 4), (9, 1)}, what is the range? 10. Completely factor the polynomial. 4x^2 + 20x + 25 11. Find the product. (2x^2 - 4x + 1)(5x - 7) 12. Solve for x. |2x - 8| > 6 13. Find the product. (25x^2 + 20xy + 16y^2)(5x - 4y) 14. True or False If F = {(2, 5), (3, 2) (4, 6), (5, 1), (7, 2)}, then F is a function. 15. Find the difference. (4x^2 - 6x + 8) - (3x^2 - 6x + 2) 16. If f(x) = 2x^2 + 3, find f(3). 17. What is the slope of the line 4x + 6y = 12? 18. Solve for x. x/4 + 6 = 10 19. Completely factor the polynomial. 100x^2 - 25 20. A line passes through the point (-3, -3) and has a slope of 1/2. What is the equation of the line? 21. If x varies directly with y and x = 2.5 when y = 10, find x when y = 16. 22. Simplify. x^3 / x^5 23. Solve for x. 2(x - 4) = 6(x + 2) 24. If x varies inversely with y and x = 4 when y = 8, find x when y = 16. 25. Together, Bob and Tom take 4 hours to get the yard work done. If Bob works alone, it takes him 6 hours. How long would it take Tom, working alone, to do all of the yard work?
Algebra 1 help.. Please?
1. 6th degree binomial (D) 2. x^4y^2 + 4x^2y + 8x (B) 3. 6x^3 - 8x - 5 + 3x^3 + 6x + 2 = 9x^3 - 2x - 3 (A) 4. product of powers (C) 5. (2x^2 y^6 z^5)(5x^4 y^5 z^3) = 10x^6 y^11 z^8 (D) 6. (5a^7b)^2 = (5a^7b)(5a^7b) = 25a^14b^2 (D) 7. (2x^7y)^2 (y^5)^3 = 4x^14y^2 y^15 = 4x^14 y^17 (D) 8. (a^9 * b^10) / (a^2 * b^7) = a^7 b^3 (D) 9. (a^3/b^7)^2 = (a^6)/(b^14) (C) 10. (4*b)/a^-10 = 4a^10b (B) 11. (-11x^3y^12z^3)^0 = 1 (D) 12. -3x^2 (5x - 4x^2 - 6) = 12x^4 - 15x^3 + 18x^2 (B) 13. (-2x + 10y)(4x - 6y) = -8x^2 + 52xy - 60y^2 (B) 14. (2x - 6)(3x^2 - 3x - 6) = 6x^3 - 24x^2 + 6x + 36 (A) 15. (6x + 2)^2 = (6x + 2)(6x + 2) = 36x^2 + 24x + 4 (D) 16. (7x - 6)^2 = (7x - 6)(7x - 6) = 49x^2 - 84x + 36 (D) 17. (6x - 2)(6x+ 2) = 36x^2 - 4 (D) 18. ?? 19. ((-5)(x^3) + 20(x^2) - 25(x)) / -5x = x^2 - 4x + 5 (C) 20. (-5x^2 - x^3 - 3) - (9x^3 + 3x - 6x^2) = -10x^3 + x^2 - 3x - 3
I'm stuck on these 3 short Algebra Q's. Please help?
1. Find the sum of the three expressions and choose the correct answer: 2x^2y^2 - 7xy + 5y^2 8xy - 3y^2 x^2y^2 + 4y^2 A)-3x^2y^2 - xy + 6y^2 B)3x^2y^2 + xy + 9y^2 C)3x^2y^2 + xy + 6y^2 D)3x^2y^2 + 15xy + y^2 2. Simplify 5m - {2n + [3p - (m + 3n - 7p)]}. A)6m - 5n - 10p B)4m + n + 10p C)6m + n - 10p 3. Find the product: 2x^3(x^2 + 5x + 9) A)2x^6 + 28x^3 B)2x^5 + 10x^4 + 18x^3 C)2x^5 + 10x^3 + 18x^3
Last questions and im done with algebra!!! please help!?
1)evaluate f/4 if f = 1/5 f/4 = (1/5)/4 = 1 / 20 2) Find the domain of y = x + 3 if the range is (-1,1,3,5) [range = y's, domain = x's] [y = x + 3] -1 = x + 3 x = -4 1 = x + 3 x = -2 3 = x + 3 x = 0 5 = x + 3 x = 2 Domain: {-4, -2, 0, 2} 3) Find the product: -5a²(a² - 6a + 7) = -5a^4 + 30a³- 35a² 4) Find the product: (6y + 5) (2y + 3) = 12y² + 18y + 10y + 15 = 12y² + 28y + 15 5) Factor the polynomial s² + 25. (s - 5)(s - 5) = No, not right = s² - 10s + 25 (s + 5)(s + 5) = No, not right = s² + 10s + 25 (s - 5)(s + 5) = No, not right = s² - 25 PRIME 6) Find the value of c to make g² - 18g + c a perfect square g² - 18g + c (g - 9)(g - 9) = (g-9)² g² - 18g + 81 c = 81 7) Solve the system of equations y = 2x - 10 and x - y = 7. x - y = 7 x - (2x - 10) = 7 x - 2x + 10 = 7 -x = -3 x = 3 3 - y = 7 -y = 4 y = -4 8) Simplify: √(75a²b³c) (use absolute value symbols if necessary). = √(25*3 a²b³c) = ± 5ab√(3bc) 9) Simplify: y² - 6y + 9 / y² - 3y = (y - 3)(y - 3) / y(y - 3) = (y - 3) / y