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Difficulty:
65%
(hard)
Question Stats:
61% (02:46) correct
39%
(02:48)
wrong
based on 612
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Given the two lines y = 2x + 5 and y = 2x - 10, what is the area of the largest circle that can be inscribed such that it is tangent to both lines
A. \(\frac{45}{4}*\pi\)
B. \(2\sqrt{45}\pi\)
C. \(27\pi\)
D. \(27\sqrt{2}\pi\)
E. \(45\pi\)
A. \(\frac{45}{4}*\pi\)
B. \(2\sqrt{45}\pi\)
C. \(27\pi\)
D. \(27\sqrt{2}\pi\)
E. \(45\pi\)
Updated on: 26 Jun 2024, 05:21
Originally posted by BrushMyQuant on 02 Sep 2012, 20:17.
Last edited by BrushMyQuant on 26 Jun 2024, 05:21, edited 11 times in total.
Last edited by BrushMyQuant on 26 Jun 2024, 05:21, edited 11 times in total.
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↧↧↧ Detailed Video Explanation to the Problem ↧↧↧
Given the two lines y = 2x + 5 and y = 2x - 10, what is the area of the largest circle that can be inscribed such that it is tangent to both lines
Diameter of the circle will be equal to the distance between these two parallel tangents.
Distance between parallel lines is given by the formula
d = \(\frac{|C_1–C_2|}{\sqrt{}(A^2+B^2)}\)
Here \(C_1\) = 5, \(C_2\)=-10
A = 2
B = -1
So, d = \(\frac{|5–(-10)|}{\sqrt{}(2^2+(-1)^2)}\) = \(\frac{|15|}{\sqrt{}5}\) = \(\frac{3 * 5}{\sqrt{}5}\)
=> d = \(3\sqrt{}5\)
=> Area = \(\frac{(\pi * d^2)}{4}\)
= \(\pi * \frac{(3\sqrt{}5)^2}{ 4}\)
= \(\frac{45 \pi}{4}\)
So, Answer will be A
Hope it helps!
Watch the following video to Learn Basics of Circles
Given the two lines y = 2x + 5 and y = 2x - 10, what is the area of the largest circle that can be inscribed such that it is tangent to both lines
Diameter of the circle will be equal to the distance between these two parallel tangents.
Distance between parallel lines is given by the formula
d = \(\frac{|C_1–C_2|}{\sqrt{}(A^2+B^2)}\)
Here \(C_1\) = 5, \(C_2\)=-10
A = 2
B = -1
So, d = \(\frac{|5–(-10)|}{\sqrt{}(2^2+(-1)^2)}\) = \(\frac{|15|}{\sqrt{}5}\) = \(\frac{3 * 5}{\sqrt{}5}\)
=> d = \(3\sqrt{}5\)
=> Area = \(\frac{(\pi * d^2)}{4}\)
= \(\pi * \frac{(3\sqrt{}5)^2}{ 4}\)
= \(\frac{45 \pi}{4}\)
So, Answer will be A
Hope it helps!
Watch the following video to Learn Basics of Circles
29 Sep 2012, 14:23
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dvinoth86
Even if you don't know the formula for the distance between two parallel lines, you can figure it out.
The two lines are parallel, having the same slope 2.
A circle tangent to both lines will have the diameter equal to the distance between the two lines. To find this distance,
take the perpendicular from the origin to each of the lines.
The origin and the x-intercept and the y-intercept of the line \(y = 2x - 10\) form a right triangle with legs 5 and 10 and hypotenuse \(5\sqrt{5}\).
Therefore, the height corresponding to the hypotenuse is \(10\cdot{5}/5\sqrt{5}=2\sqrt{5}\) (use the formula height = leg*leg/hypotenuse).
The distance between the origin and the line \(y = 2x + 5\) is half of the distance between the origin and the line \(y = 2x - 10\), because the x-intercept and the y-intercept of the line \(y = 2x + 5\) are -5/2 and 5, which with the origin, form a right triangle similar to the right triangle discussed above.
So, the diameter of the circle is \(3\sqrt{5}\) and the area of the circle is \(\pi\cdot9\cdot{5}/4=(45/4)\pi\).
