Triangle PQR where P(1,4), Q(-3,-4),R(7,k) is right-angled at Q. find the value of k

k = - 9

Step-by-step explanation:

Since the triangle is right- angled at Q then PQ is perpendicular to RQ

Calculate the slope of PQ using the slope formula

m =

with (x₁, y₁ ) = P(1, 4) and (x₂, y₂ ) = Q(- 3, - 4)

= = = 2

Given a line with slope m then the slope of a line perpendicular to it is

= - = - , thus

= -

Calculate the slope of RQ and equate to -

= = - , that is

= - ( multiply both sides by 10 )

k + 4 = - 5 ( subtract 4 from both sides )

k = - 9

The parabola has a horizontal tangent line at the point (2,4)

The parabola has a vertical tangent line at the point (1,5)

Step-by-step explanation:

Ir order to perform the implicit differentiation, you have to differentiate with respect to x. Then, you have to use the conditions for horizontal and vertical tangent lines.

-To obtain horizontal tangent lines, the condition is:

(The slope is zero)

--To obtain vertical tangent lines, the condition is:

(The slope is undefined, therefore the denominator is set to zero)

Derivating respect to x:

=

Solving for dy/dx:

Applying the first conditon (slope is zero)

Solving for y (Adding 2x+4, dividing by 2)

y=x+2 (I)

Replacing (I) in the given equation:

Replacing it in (I)

y=(2)+2

y=4

Therefore, the parabola has a horizontal tangent line at the point (2,4)

Applying the second condition (slope is undefined where denominator is zero)

2y-2x-8=0

Adding 2x+8 both sides and dividing by 2:

y=x+4(II)

Replacing (II) in the given equation:

Replacing it in (II)

y=1+4

y=5

The parabola has vertical tangent lines at the point (1,5)