Dr. J's Maths.com
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 Identification of tangents etc. 1. 2. 3. Given that the line lx + my + n = 0 is a tangent to x2 = 4ay, prove . 4. 5. A parabola whose equation is y = kx2 (where k is a constant) has the line y = -6x + 3 as a tangent. (i) By equating the two given equations, find a quadratic equation in terms of x and k. (ii) By using the discriminant of the quadratic equation found in (i), find the value of k. (iii) Find the coordinates of the point of contact between the tangent and the parabola. (iv) Sketch the parabola and the tangent line, showing the coordinates of the point of contact and where the tangent line cuts the x and y axes. Answer.(i) kx2 + 6x - 3 = 0(ii) k = -3(iii) Point of contact (1, -3). 6 Nature of roots. 15. Show that the constants k and l have different signs if the quadratic equation has no real roots . 16. (i) For what values of k does the quadratic equation kx2 + (k + 3)x - 1 = 0 have real roots? (ii) Why cant the equation ever be positive definite? Show that the quadratic equation (a2 - b2)x2 + 2b(a - c)x + (b2 - c2) = 0 has real and rational roots for all values of x if a, b and c are rational.