Use implicit differentiation to find an equation of the tangent line to the curve at the given point.x^2 + y^2 = (3x^2 + 4y^2 ? x)^2(0, 0.25)(cardioid)y=?

Answer:y = 2(x - 1/4)Step-by-step explanation:To find an equation of the tangent to a given curve, we need two vital information; one is the slope of the tangent (SOP) and the other is the point of tangency (POT).to find the slope of tangent i will use the implicit differentiation to find the slope then apply the point given so, first, the slope using implicit differentiation;f(x) = [tex]x^{2} +y^{2} =(3x^{2}+4y^{2} -x) ^{2}[/tex]f'(x) = {tex}2x + 2y dy/dx = 2(3x^{2} + 4y^{2} - x)(6x + 8y dy/dx - 1){/tex}applying the co-ordinates given; x = 0 and y = 1/4SOP  2(0) + 2(1/4) dy/dx = 2[3(0)^2 + 4(1/4)^2 - 0][6(0) + 8(1/4) dy/dx - 1)1/2 dy/dx = 2[4(1/16)][(8/4) dy/dx - 1]1/2 dy/dx = 2[1/2][2 dy/dx - 1]1/2 dy/dx = 2 dy/dx - 1putting the dy/dx together 1 = 2 dy/dx - 1/2 dy/dx 1 = dy/dx (2 - 1/2)make dy/dx subject 1 / 1/2 = dy/dx therefore SOP = 2POT f(x) = (0)^2 + (1/4)^2 = [3(0)^2 + 4(1/4)^2 - 0]^21/16 = [4(1/16)]^21/16 = (1/4)^21/16 = 1/16y = 0using the slope formulam = y - y2/ x - x22 = y - 0/ x - 1/4y = 2(x - 1/4)