Geometry Answers

1. A statuary hall is elliptical in shape. It measures 42 feet wide and 96 feet long. If a person is standing at one focus, her whisper can be heard by a person standing at the other focus. How far apart are the two people?



2. A conic section is given by the equation 57x^2+14√3 xy+43y^2-576=0, identity and sketch the conics.


3. Find the area of the region inside both the rose curve r=sin⁡〖(2θ)〗 and the circle r=cos⁡θ. Choose the most appropriate graph and intervals to the best of your interest to solve this question


4. Sketch and identify the curve defined by the parametric equations


y=t+1, x=t^2-2t , -4≤t≤6


5. Identify and sketch the curve 9x^2-4y^2-72x+8y+176=0 using the appropriate graph. Find the equations of the asymptotes.

1. A door of a lecture hall is in a parabolic shape. The door is 56 inches across at the bottom of the door and parallel to the floor and 32 inches high. Sketch and find the equation describing the shape of the door. If you are 22 inches tall, how far must you stand from the edge of the door to keep from hitting your head?

1. A door of a lecture hall is in a parabolic shape. The door is 56 inches across at the bottom of the door and parallel to the floor and 32 inches high. Sketch and find the equation describing the shape of the door. If you are 22 inches tall, how far must you stand from the edge of the door to keep from hitting your head?

f a chord of the parabola y^2 = 4ax is a normal at one of its ends, show that its mid-point lies on the curve 2(x − 2a) = y^2/a + 8a^3/y^2

Verify that the point P(acosθ,bsinθ) lies on the ellipse

x^2/ a^2+ y^2/ b^2= 1,

where a and b are the semi-major and semi-minor axes respectively of the ellipse . Find the gradient of the tangent to the curve at P and show that the equation of the normal at P is axsinθ−by cosθ = (a^2 −b^2)sinθ cosθ. If P is not on the axes and if the normal at P passes through the point B(0,b), Show that a^2 > 2b^2. If further, the tangent at P meets the y-axis at Q, show that

|BQ| =

a^2/ b^2

.

2. Verify that the point P(a cos θ, b sin θ) lies on the ellipse

x^2/a^2 + y^2/b^2 = 1,

where a and b are the semi-major and semi-minor axes respectively of the ellipse .

Find thegradient of the tangent to the curve at P and show that the equation of the normal at P is

ax sin θ − by cos θ = (a^2 − b^2) sin θ cos θ.


If P is not on the axes and if the normal at P passes through the point B(0, b), Show that a^2 > 2b^2.


If further, the tangent at P meets the y-axis at Q, show that

|BQ| =a^2/b^2 .

1. (a) If a chord of the parabola y^2 = 4ax is a normal at one of its ends, show that its mid-point lies on the curve 2(x − 2a) = y^2/a + 8a^3/y^2

.

Prove that the shortest length of such a chord is 6a√3.


(b) Find the asymptotes of the hyperbola

x^2 − y^2 + 2x + y + 9 = 0.

Line PR is perpendicular to line QS, angle QPR=60°, angle PSR=30° and line PQ=8cm. Calculate line SR correct to two significant figures

A learner in your class is confused about the concepts of perimeter and area. Use

examples to explain and illustrate the difference between these concepts.

Mpho has a circular swimming pool with a radius of 6.2 m. She wants to buy net to

cover the pool completely. What is the minimum length and width of the piece of net

that she must buy to cover the pool?