Question

Show that the curve x = 7 cos(t), y = 5 sin(t) cos(t) has two tangents at (0, 0) and find their equations.

y = __________ (smaller slope)
y = __________ (larger slope)

Answer 1

y= 5 y =7 thats the smaller slip and larger

Answer 2

The equations of the two tangents are; Larger slope;  y = (5/7)x Smaller slope;  y = (-5/7)x

What is the slope?

Slope tells how vertical a line is.

The more the slope is, the more the line is vertical. When slope is zero, the line is horizontal.

To find the slope, we take the ratio of how much the line's height increases as we go forward or backward on the horizontal axis.

This is because the more the height of the line to thee amount we walk or run on the horizontal axis, the more the slope is. Thats why we took difference of horizontal axis in denominator and difference of vertical axis on numerator.

The curve x = 7 cos(t), y = 5 sin(t) cos(t) has two tangents at (0, 0) and we need to find their equations.

We can write

 cos(t) = x/7

so

 y = 5(±√(1 -(x/7)^2)*x/7

Then

 y' = (±5/7)*(√(1 -(x/7)^2) + x/(2√(1 -(x/7)^2)*(-2(x/7))

The limit as x → 0 is ±5/7

The equations of the two tangents are;

Larger slope;  y = (5/7)x

Smaller slope;  y = (-5/7)x

Learn more about slope here:

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