If P(a, a²) is any point on the parabola y = x², except for the origin, let Q be the point where the normal line intersects the parabola again. Show that the line segment PQ has the shortest possible length when a = 1//2.

Q: Find an equation of the normal line to the parabola y= x2 -5x +4that is parallel to the line  x-3y…

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Q: At what point does the normal to y = 2 + 2x + 2x² at (1,6) intersect the parabola a second time?

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Q: What is the shortest distance from the curve x^2+xy+y^2=16 to the origin?

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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: At what point does the normal to y=4x2−(2+3x)y=4x2−(2+3x) at (1,−1)(1,−1) intersect the parabola a…

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Q: Find the condition that the line ax+by=1 may be normal to the curve an-1y=xn

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Q: 13) Find the points of intersecti on of the parabola and the line. y=2x^2-3x-1 and y=x+5

A: Equate the equations with each other and solve for x to find the point of intersection .

Q: Find the length of the curve y x² Inx 8 at (2 <x< 4)? -

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

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Q: For what values of a and b is the line 2x + y = b tangent to the parabola y ax? when x = 2?

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Q: Find the length of the curve x = 5-t, for 1 < t < 2. length = y = 3-t, z = 3t 5,

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Q: Find the exact length of the curve. 1 y = 4 -x² - In(x), 1< x < 8 2

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Q: Find the condition that the line ax+by =1 may be normal to the curve an-1y =xn

A: Given thatan-1y =xn Solution is as follows

Q: Show that the normal line at any point on the circle x2+ y2 = r2  passes through the origin.

A: Consider the given equation of circle as x2+y2=r2

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

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Q: Find an equation of the normal line to the parabola y = x2 - 5x + 4 that is parallel to the line x -…

A: y=x2-5x+4 Differentiating with respect to x, we get dydx=2x-5 Now slope of the normal line is,…

Q: y = 4 1 ,2 In x, 1<x < 2 2 -

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Q: Find the length of the curve from the entire hypocycloid 2 2 x3 + Y³ = a³.

A: Explained below

Q: If the line r = (2-t, -2 + 13t, 8t) is the normal tangent line to the level surface axyz²by² = -14…

A: answer is in next step . please give a like !!

Q: What is the maximum vertical distance between the line  y = x + 12  and the parabola  y = x2 for…

A: See the attachment

Q: Find the exact length of the curve. 는x - - In(x), 1<x<6 4 2

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Q: Find the exact length of the curve. 36y2 = (x² – 4)³, 4 < x < 9,

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Q: Where does the normal line to the parabola y= x- x2 at  the point (1, 0) intersect the parabola a…

A: Normal to a curve is a line which is perpendicular to the curve at its point of intersection with…

Q: Find the length of the curve y = In (1 – 2*) over the interval [-,].

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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 an equation of the normal line to the parabola y = x 2 - 5 x + 4 that is parallel to the line…

A: Given:       y=x2-5x+4 is parallel to the line x-3y=5.

Q: Find the 0lar eauatiDo for the curve represented by the given Cartesian euation.

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Q: The shortest distance between the parabolas y² = x – 1 and x' = y-l is V2, then the value of…

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

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

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Q: What is the maximum vertical distance between the line y = x + 42 and the parabola y = x for -6 s xs…

A: Consider the equation y=x+42 and y=x2; -6≤x≤7. Substitute y=x+42 in y=x2.…

Q: 1. Find the shortest distance from the point (1,0, –2) to the plane x+2y+z = 4.

A: We have to find shortest distance

Q: Find the exact length of the curve. 1 x² -In(x), 1< x < 8 4

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Q: Find the minimum distance from the parabola: x - y2 = 0 to the point (0,3).

A: x - y2 = 0 x = y2  Let us take a point (x,y)=(y2, y) on the parabola

Q: The equation of the curve in the tx-plane which passes through the point (2,3), and has, at every…

A: In this question, slope of curve and point is given and we have to find out equation of cuve.

Q: Find the shortest distance from the point (4,2) to the curve x^2 Зу^2 %3D 12 +

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Q: Show that the normal line at any point on the circle x2 + y2 = r2 passes through the origin

A: Assume that we have the circle of equation x2+y2=r2 and we want to get the normal equation at x0,y0.…

Q: Find the exact length of the curve. 1 y = -x° 4 x² -- In(x), 1<x< 4 | 2

A: Given curve, y=14x2-12lnx   ;  1≤x≤4

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

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

A: To show that the two curves are orthogonal.

Q: Find the length of the curve y Inx at (2<x< 4)?

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Q: What is the value of c such that the line y=2x+4 is tangent to the parabola y=cx^2.

A: Given that the line y=2x+4 is tangent to the parabola y=cx2. The derivative of y=cx2 is dydx=2cx.…

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

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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: Find the length of the curve defined by y = 3 In 3 from x = 8 to x = 9.

A: Use derivative of function and length of curve formula .

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

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

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Q: Find the exact length of the curve. y = In(1 - x²), o sxs

A: Find dy/dx, then put it in the arc length formula.

Q: Find the parabola y = ax2 + bx + c that passes through (2,1) and be tangent to the line y = 2x + 4…

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Unitary Method

The word “unitary” comes from the word “unit”, which means a single and complete entity. In this method, we find the value of a unit product from the given number of products, and then we solve for the other number of products. 

Speed, Time, and Distance

Imagine you and 3 of your friends are planning to go to the playground at 6 in the evening. Your house is one mile away from the playground and one of your friends named Jim must start at 5 pm to reach the playground by walk. The other two friends are 3 miles away.

Profit and Loss

The amount earned or lost on the sale of one or more items is referred to as the profit or loss on that item.

Units and Measurements

Measurements and comparisons are the foundation of science and engineering. We, therefore, need rules that tell us how things are measured and compared. For these measurements and comparisons, we perform certain experiments, and we will need the experiments to set up the devices.