Show that #CM# and #RQ# are perpendicular ?

Answer 1

See below.

Given #R=(3a,a)# and #Q=(a,-3a)# the mid point of #RQ# is given by

#M=1/2(R+Q)=(2a,-a)#

The circle's center is #C=(0,0)#

so #CM=(2a,-a)-(0,0)=(2a,-a)# and

#RQ = R-Q =(2a,4a) #

Finally

#RQ cdot CM = (2a,4a) cdot (2a,-a) = 4a^2-4a^2=0#

Attached a figure

Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
Answer 2
The x coordinate of the chord goes from #3a# to #a#, therefore, the x coordinate of the midpoint is halfway between them, 2a.
The y coordinate of the chord goes from #a# to #-3a#, therefore, the y coordinate of the midpoint is halfway between them, -a.
The coordinates of the midpoint are #M(2a,-a)#
The slope, #m_1#, of the chord RQ is:
#m_1 = (-3a - a)/(a - 3a)#
#m_1 = (-4a)/( -2a)#
#m_1 = 2#
The equation of the circle #x^2 + y^2 = 10a^2# is a special case of the more general equation #(x - h)^2+(y-k)^2 = r^2# where the center is the origin #(0,0)#.
The slope, #m_2#, of the line from the center to the midpoint is:
#m_2 = (-a - 0)/(2a - 0)#
#m_2 = (-a)/(2a)#
#m_2 = -1/2#

Please observe that:

#m_1m_2 = -1#

This shows that the two lines are perpendicular.

Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
Answer 3

To show that CM and RQ are perpendicular, we can use the property of perpendicular lines in coordinate geometry. If the product of the slopes of two lines is -1, then they are perpendicular. Let's denote C as (x1, y1), M as (x2, y2), R as (x3, y3), and Q as (x4, y4).

The slope of line CM, m_CM, is (y2 - y1) / (x2 - x1). The slope of line RQ, m_RQ, is (y4 - y3) / (x4 - x3).

If CM and RQ are perpendicular, then: m_CM * m_RQ = -1.

So, we need to show that: ((y2 - y1) / (x2 - x1)) * ((y4 - y3) / (x4 - x3)) = -1.

If this equation holds true, then CM and RQ are perpendicular.

Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
Answer 4

To show that CM and RQ are perpendicular, we can use the concept of slopes. Perpendicular lines have slopes that are negative reciprocals of each other.

Let's denote CM as line m and RQ as line n.

The slope of CM (m) can be calculated using the coordinates of points C and M. Similarly, the slope of RQ (n) can be calculated using the coordinates of points R and Q.

If the product of the slopes of two lines is -1, then the lines are perpendicular.

Therefore, to demonstrate that CM and RQ are perpendicular, we need to show that the product of their slopes is -1.

Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

Not the question you need?

Drag image here or click to upload

Or press Ctrl + V to paste
Answer Background
HIX Tutor
Solve ANY homework problem with a smart AI
  • 98% accuracy study help
  • Covers math, physics, chemistry, biology, and more
  • Step-by-step, in-depth guides
  • Readily available 24/7