Find the distance of the point (-2,3,-4) from the line (x+2) /3 = (2y+3) /4 = (3z+4) /5 measured parallel to the plane 4x+12y-3z+1=0.
Answers
Let us consider P = ( -2, 3, -4 )
Let π be the plane parallel to the given plane through P, and let Q be the point where the given line π. The distance required is then the length of the segment PQ.
Let us find the value of π:
We know that π is parallel to the given plane so the equation of π which differs only in the constant term.
4x + 12y – 3z + c = 0………………………(1)
Where c = some constant term
Where P is in π, its coordinates satisfy the equation
=> 4(-2) + 12(3) – 3(-4) + c = 0
=> -8 + 36 + 12 + c = 0
=> c = -40
So the equation for π is
4x + 12y – 3z – 40 = 0……………………..(2)
Let us find Q:
For points on the line (x+2)/3=(2y+3)/4
=> 4x+8 = 6y+9
=> 6y = 4x-1 (x+2)/3=(3z+4)/5
=> 5x+10 = 9z+12
=> 9z = 5x-2
For points in plane π
4x + 12y – 3z – 40 = 0
=> 12x + 36y – 9z – 120 = 0
Point Q is on both line and π
12x + 36y – 9z – 120 = 0
=> 12x + 6(4x-1) – (5x-2) – 120 = 0
=> 12x + 24x – 6 -5x + 2 – 120 = 0
=> 31x = 124
=> x = 4
Then y = (4x-1)/6
y = 15/6
y = 5/2
z = (5x-2)/9
z = 18/9
z = 2
Thus Q = (4, 5/2, 2)
Let us calculate the length of PQ:
We know P = ( -2, 3, -4 ) and Q = ( 4, 5/2, 2 )
Length of PQ = √((-2 – 4)² + (3 – 5/2)² + (-4 – 2)²)
Length of PQ = √((-6)² + (1/2)² + (-6)²)
Length of PQ = √(36 + 1/4 + 36)
Length of PQ = √(72 + 1/4)
Length of PQ = √(288/4 + 1/4)
Length of PQ = √(289 / 4)
Length of PQ = 17 / 2
The distance between coordinates (4,5/2,2) and (2,3,−4) is 17/2 unit.
(i) Given: Each radius of a circle is a chord of the circle
The radius meets the circle at only one point whereas a chord meets the circle at two different points.
Any radius of the circle cannot be the chord of the circle.
∴ The given statement is false.
(ii) Given: The centre of a circle bisects each chord of the circle
only the diameter of a circle is a chord at which the centre of the circle exists.
The centre does not bisect all chords.
∴ The given statement is false
(iii) Given: a circle is a particular case of an ellipse.
Let us consider the equation of an eclipse
x2/a2 + y2/b2 = 1
If a=b then
x2 + y2 = 1
This is an equation of circle.
So the circle is a particular case of an eclipse.
∴ The given statement is true.
(iv) Given: if x and y are integers such that x > y then -x < -y
Where x > y then by the equation of inequality
⇒ -x < -y
∴ The given statement is true.
(v) Given: √11 is a rational number.
Every rational number can be expressed in the form P/P where p and q are integers and q≠0.
But √11cannot be expressed in the form of p/q.
∴ The given statement is false.