Prove algebraically that the straight line with equation x = 2y + 5 is a tangent to the circle with equation x^(2) + y^(2) = 5
If a line is a tangent to a circle, it will have exactly one point where the line intersects the circle. Therefore, when we set the equation of the straight line equal to the equation of the circle, we should have only on set of solutions. If x = 2y + 5 and x2 +y2 =5, then we can substitute x2 for (2y +5)2..Then we get (2y +5)2 + y2 = 5. Expand and Simplify the equation to get: 4y2 +20y + 25 + y2 = 5When we collect like terms, this can be simplifed to : 5y2 + 20y + 20 = 0As all the terms are a multiple of 5, and the equation is set to zero, we can divide through by 5 and get the following equation:y2 + 4y + 4 = 0As we want to show that this equation has only one solution, we can use the discriminant, b2 - 4ac. If this is equal to zero, then we know that the equation has only one solution. so b2 - 4ac = (4)2 - 4(1)(4) = 16 - 16 = 0. As the discriminant is equal to zero, we can say the there is only one solution of the equation so the line has to be a tangent to the circle.
