(a) Prove that any two distinct tangent lines to a parabola intersect. (b) Demonstrate the result of part (a) by finding the point of intersection of the tangent lines to the parabola x2 − 4x − 4y = 0 at the points (0, 0) and (6, 3).

Q: Show that the equation of the tangent line to the hyperbola                        x2/a2 − y2/b2 = 1…

A: Given Data The equation of the hyperbola is x²/ a²- y²/ b²=1. The points are (x0,y0).…

Q: 7-18 - Find the exact length of the curve. 7. y = 1 + 6x³/², 0 < x < 1

A: We find y' using the power rule

Q: 8-1) Assume that two surfaces to create a curve : 炉 x² + x²-x²3² = -7 are intersectiong 3x₁² + x₂ +…

A: Introduction: In three dimensional Euclidean geometry, the directional derivative of a surface is…

Q: Find the equations of the tangent line in the given plane. Tangent to the parabola z = x² + 3y² , x…

A: Tangent line equation problem....

Q: Find an equation of the tangent plane to the given surface at the specified point. 2. (-16, 4, -1)

A:

Q: 1. Let P(1, yı) be a point on the parabola r = 4py, p> 0, with focus F(0, p). Let a be the angle…

A:

Q: Show that line d:3x-4y-5=0 is tangent to hyperbola H:x2-4y2=5 and find the point of contact of the…

A:

Q: Find all hyperbolic lines that passes through i +1 and parallel to the y-axis.

A: Find all hyperbolic lines that passes through i+1 and parallelto the y-axis.

Q: Find points P and Q on the parabola y = 1 – x² so that the triangle ABC formed by the x- axis and…

A: We have △ABC is an isosceles right angle triangle. So the interior angle at B and C must be 45° The…

Q: 1 - Sketch the curve : y=-(x3– 6x²+9x+6). 6 y=Lcx³-

A:

Q: On a separate piece of paper, sketch the graph of the parabola y = x² + 9. On the same graph, plot…

A: Sketch the figure.

Q: Find the equations of the tangent line in the given plane. Tangent to the parabola z =x^2 +3y^2,…

A: z=x2+3y2 , x=1 at the point (1,2,13)

Q: Find constants a and b so that the minimum for the parabola f(x) = x2 + ax + b Occurs at the point…

A: The given parabola is : f(x)=x2+ax+b We need to find the values of a and b so that the minimum for…

Q: 3. Determine the orthogonal trajectory of the family of parabolas x² – y + 32x = ketch some of the…

A: The given parabola is x2-y+32x=0 ........................ (1)  Differentiate with respect to x,

Q: Find the equation of the tangent to the hyperbola x² – 3y² = 1 at the point S(2,-1). A) 2x + 3y – 1…

A:

Q: 3. Find an equation of the tangent plane to the paraboloid defined by z= 5x +4y? at the point…

A: The given equation is; z=5x2+4y2 at the point (-1,1,9).

Q: 2 2 the equation of the tangent plane to the surface x² + y² + y² +2²= -1 at the point (2, -3, 4)

A: I am attaching image so that you understand each and every step.

Q: Derive the orthogonal trajectories of the family of hyperbolas ??=?.

A:

Q: Prove that the product of the perpendicular distances F1R1 and F2R2 from the foci to a tangent line…

A:

Q: The plane x = 1 intersects the paraboloid z = x2 + y2 in a parabola. Find the slope of the tangent…

A: Given that the plane x = 1 intersects the parabolic z = x^2 + y^2 in a parabola.   So,…

Q: Determine the equation of the tangent and normal to : 1. The parabola y² = 6x - 3 perpendicular line…

A:

Q: 1. Let P(r1, y1) be a point on the parabola r? be the angle between the parabola and the line…

A: angle with a curve at a point means angle with tangent line at that point of curve .

Q: Prove that the curves y2 = 4x and y2-6x+1 = 0 touch each other at the point (1,2)

A:

Q: a) Determine the point at which the slope of the tangent to each parabola is zero. i) y= 6x2-3x + 4…

A:

Q: Prove that the curves y2 = 4x and x2+ y2-6x+1 = 0 touch each other at the point (1,2)

A:

Q: Find the equation of the tangent line to right branch of the hyperbola π π x = sect, y = tant, ≤t≤at…

A: Given :x=secty=tant

Q: Find the equation of the tangent line to right branch of the hyperbola x = sec t, y tant, -sts at…

A: Given query is to find the equation of tangent line.

Q: (a) Find the first order DE corresponding to the family of parabolas y = Cx² . (b) Find the equation…

A: a)The family of the curve of the parabola is y=Cx2 To find the first order differential equation…

Q: Find constants a and b so that the minimum for the parabola f(x) = x + ax + boccurs at the point…

A:

Q: Find an equation of the tangent line to the hyperbola defined by 5x^(2)+4xy−2y^(2)−4x=19 at the…

A:

Q: Show that the equation of the tangent line to the hyperbola x2/a2 − y2/b2 = 1 at (x0, y0) is x0x/a2…

A:

Q: Find the equation of the curve, in the XY plane, that passes through the point (2, 2) and is such…

A: Differential equation

Q: 22=xy represents the origin. ) Prove that the equation with vertex at

A:

Q: The points P(ap2 , 2ap) and Q(aq2 , 2aq) lie on the parabola y2= 4ax. Prove that if PQ is a focal…

A:

Q: If a parabola defined by f(x)=2−x^2 is inscribed under an equilateral triangle ABC, so that the…

A: Given data:     The given equation of parabola is: f(x)=2-x2.     The below figure represents the…

Q: 1. Find the equation of the tangent lines to the ellipse x² + 4y = 8 perpendicular to the line x +…

A:

Q: The semi-hyperbola H(X)=(sqrt(x^2-1)) is continous at which x interval?

A:

Q: A parabola has an equation of y²=2x, compute the radius of curvature at point (2, 2)

A:

Q: Exer. 35-36: Find equations of the tangent line and normal line to the hyperbola at the point P. 35…

A: As per guideline we have to solve one question

Q: The tangent and the normal to the parabola y²=20x at the point T(5,10) meet the x-axis at the points…

A: Details solution is given below:

Q: 1 Let P be the parabola with equation y Lx2, L be the line with equation y = V3 7. 3 1 and M be the…

A: Note: There are multiple sub questions. No specific question has been asked. So, We will solve the…

Q: 4. Classify the type of the equation as elliptic, parabolic or hyperbolic. Uxx xuy+xu 0 =

A: The given problem is to classify the given partial differential equation as elliptic, parabolic or…

Q: 1. Let P(x1, Y1) be a point on the parabola x?: be the angle between the parabola and the line…

A:

Q: 6. Show that if m is any real number, then there are exactly two lines of slope m that are tangent…

A:

Q: On a separate piece of paper, sketch the graph of the parabola f(T)= x+ 8. On the same set of axes,…

A: The equation of parabola is fx=x2+8 The coordinate of the point from which the tangents are drawn on…

Q: 3 On a separate piece of paper, sketch the graph of the parabola f(x) = x² + 6. axes, plot the point…

A: Given

Q: 2. Find the equation of the tangent line to the parabola 2y^2+36x=0 which is parallel to the line…

A:

Q: 16. Evaluate (x + V9) ds where C consists of two segments: the part of the parabola y = x² from…

A: In the given question we have to evaluate the integral ∫C(x+y)ds where C is the regionconsist of two…