4. Let P(x1, y1) be a point (not the vertex) on the parabola y² = 2px. Find the equation of the line normal to the parabola at this point. Show that this normal line intersects the x axis at the point Q (x1 + p, 0).

Q: 4. Find the equations of the line tangent to the intersection of the surfaces z = x´ + 2y and

A:

Q: Prove that for any point (a, b) ≠ (0, 0) on the parabola y = x2 , the normal line intersects the…

A: The given equation of parabola is y=x2  For the point (a,b)≠ (0, 0), the slope of normal is and at…

Q: Find an equation for the set of points in an xy-plane that are equidistant from the point P and the…

A: H

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

A: The length of the curve is calculated from the graph.

Q: The tangent of the parabola x = y² + y + 1 at the point (1,0) is

A:

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: Consider the line L(t) = (4 – 3t, 17 + 4t). Then L intersects: 1. The x-axis at the point when t =…

A: Lt=4, -3t, 17+4t1  On the x‐axis , y co‐ordinate in zero       Therefore Lt will intersect y…

Q: Find an equation of the normal line to the parabola y = x – 9x + 5 that is parallel to the line x -…

A:

Q: 4. Determine the slope of the common tangent to the parabolas y=x² +1 and y=-x² - 1. y=x*2+1 (a, b)…

A:

Q: Find the exact length of the curve.   y2 = 4(x + 4)3 , 0 ≤ x≤ 2 , y&gt; 0

A: Given: y2=4x+43,0≤x≤2,y&gt;0 To Find: Find the Exact length of the curve. Concept: Length of the…

Q: Find an equation of the normal line to the parabola  y = x2 − 4x + 1  that is parallel to the line…

A:

Q: Find the point(s) of intersection of the parabola y =x2+ 2 andthe line y = x + 4.

A: The given functions are y=x2+2   and   y=x+4

Q: Determine shortest distance of the line 2x-3y-8 to the vertex of the curve x2 - 4a 4y +4 0. O 2 O…

A: We have given a line ,and find the shortest distance from vertex of parabola to line.

Q: 3. Find the line normal to the ellipse at X=-1 y= 2+4x-x3

A: Given, y=2+4x-x3 At x= -1 y=2+4-1--13y=2-4+1y=-1 Now point -1,-1

Q: At what point does the normal to y = 3x – (5 + 4x) at (1, –6) intersect the parabola a second time?…

A:

Q: Consider the line L(t) = (4 – 4t, 17 + 3t). Then L intersects: 1. The x-axis at the point when t =…

A: Consider the line L(t) = &lt;x(t) ,y(t) &gt;, then L intersect at the x-axis at the point &lt;x(t)…

Q: Find an equation of the normal line to the parabola y = x2 – 8x + 4 that is parallel to the line x –…

A: Let the equation of the normal line in the slope-intercept form is, y=mx+c where m is the slower and…

Q: Find the area of the segment cut off from the parabola x2 = 8y by the line x- 2y + 8 = U. %3D

A: Introduction ::

Q: Find the tangent lines as directed. To the parabola y=6x-3 perpendicular to the line x+3y=7

A:

Q: 7. For whit value of b will the line y =- y 3x +4x -1? 2x+ b be tangent to the parabola

A:

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

A:

Q: Find the veeton equation of a Line L going thagh the P= C-6,2) $ paeallel to the ine point %3D…

A:

Q: Find the equation of the normal line to the parabola y = x + 3x – 2, that is also parallel to 7x…

A:

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

A:

Q: What is the equation of the normal line to the ellipse 8x^2 + 32y^2 = 256 at the point (4,2)?

A: the equation of the normal line to the ellipse can be found by differentiating the given equation…

Q: 4. Find the two values of a for which the line connecting the point (1, –1) to a point (a, a²) on…

A:

Q: Find the equation of the normal line to the parabola y=x^2+2x−3, that is also parallel to −2x+2y=9

A: Given that normal line to parabola is parallel to -2x+2y=9. Slope of the line -2x+2y=9 is -…

Q: Determine the equation of the normal line to the hyperbola x - 4y- = 9 at (5, 2). 8x + 5y – 50 = 0…

A: Given equation of hyperbola is  x2-4y2=9          ...(1) We have to find the equation of the normal…

Q: 4) Suppose that the level curves of a function z = f (x, y) consists of straight lines. Must the…

A: Given that level curve of a function z=f(x,y) consists of straight lines.

Q: Find the point on the graph of z = x2 + y2 + 10 nearest the plane x + 2y - z = 0.

A:

Q: Show that the curve In (2X-1) , n> i has no stationary point

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Q: Find the shortest distance, d, from the point (10, 0, –6) to the plane x + y + z = 6. d =

A:

Q: Find an equation of the normal line to the parabola y = x2 - 8x + 9 that is parallel to the line x -…

A:

Q: Show that the curves 2x2 + 3y2 = 5 and y2 = x3 are orthogonal.

A: To show that the two curves are orthogonal.

Q: Find an equation for the set of points in an xy-plane that are equidistant from the point P and the…

A: Equidistant points mean that the distance of two points represented by P1 and P2 is equal to the…

Q: Find the points on the curve y-9x closest to the point (0, 4). ) and (O

A:

Q: Find a parabola with equction Y=ax +bx+c that has a Slope I at x -1, slope 9 at x=-1, and passes…

A:

Q: Find the minimum distance from the point (1, 1, 11) to the paraboloid given by the equation z = x² +…

A:

Q: Consider the parabola f (x) = x² – 4x and the line g(x) = –2x – 1. Determine the %3D point(s) of…

A: Given parabola,y=x2-4x                     (1)Given line y=-2x-1                     (2)Put value of…

Q: Determine the equation of the normal line to the hyperbola x^2 - 4y^2 = 9 at (5,2)

A: We have to find equation of normal

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

A:

Q: Find the equation of tangent line to the parabola y² = 6x – 3 which is perpendicular to the line x +…

A:

Q: What is the equation of the normal line to the ellipse 8x^2 + 32y^2 = 256 at the point (4,2)? O y =…

A:

Q: Determine the equation of the normal line to the hyperbola x2 – 4y2 = 9 at (5, 2).

A:

Q: Find a number k such that the line y=6x+4 is tangent to the parabola y= x2 + k.

A:

Q: Find the tangent line to the ellipse x^2+4y^2=8 parallel to the line x+2y=6.

A:

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

A:

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

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

Q: (a) Prove that the line 2x 3y + 26 =0 is a tangent to the curve x + y2 - 4x + 6y – 104 = 0 hence…

A: