Circle and chord

Geometry Level 4

If the values of a a for which the common chord of circles x 2 + y 2 = 8 x^2+y^2=8 and ( x − a ) 2 + y 2 = 8 (x-a)^2+y^2=8 subtends right angle at the origin are a 1 , a 2 , … , a n , a_{1},a_{2},\ldots ,a_{n}, then find n + ∑ j = 1 n a j + ∏ j = 1 n a j n+\displaystyle \sum^{n}_{j=1}a_{j}+\displaystyle \prod^{n}_{j=1}a_{j} .


The answer is -14.

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Akshat Sharda by Akshat Sharda , 21, India

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4 solutions

Prakhar Bindal
Aug 31, 2016

An easy one.

common chord x = a/2

Homogenize x^2 +y^2 = 8 by putting 2x/a = 1 on RHS Like

x^2 + y^2 = 8(2x/a)^2

Now as they are perpendicular coefficient of x^2 + coefficient of y^2 = 0

3 Helpful 0 Interesting 0 Brilliant 0 Confused

The common chord that subtends right angle at the origin for first circle must make 45 degrees to x-axis. The rest is shown in the diagram.

2 Helpful 0 Interesting 0 Brilliant 0 Confused
Wei Chen
Aug 30, 2016

Since the two circles have same radius, and the common chord subtends 90 degrees at one of the centers, the quadrilateral formed by the two centers and two intersection points has to be a square. Thus ∣ a ∣ |a| equals length of diagonal of the square, which is

2 ⋅ r = 2 ⋅ 8 = 4 \sqrt{2}\cdot r=\sqrt{2}\cdot \sqrt{8}=4

From which we get n = 2 n=2 , a 1 = 4 a_{1}=4 and a 2 = − 4 a_{2}=-4

2 Helpful 0 Interesting 0 Brilliant 0 Confused
Kushal Bose
Aug 30, 2016

(1) Subtract two equations of circles which will give a linear equation in x.This equation is the equation of common chord.

(2) Put this equation in any one of the equation of circle and solve.This will give the two end points of the common chord.

(3) Now join these two end points with origin and measures slopes of each.

(4)Multiply the slopes which is equal to -1 as they are perpendicular to each other.

(5)On solving you will get two solutions a = 4 , − 4 a=4,-4

2 Helpful 0 Interesting 0 Brilliant 0 Confused

Instead of solving the common chord with one of the circles you can homogenize x 2 + y 2 − 8 = 0 x^2+y^2-8=0 with the common chord, which is x = a 2 x=\frac{a}{2} to directly obtain pair of straight lines intersecting at origin.

Akshat Sharda - 4 years, 9 months ago

if any one don't want to homogenize then let angle be thetha and find the length of the chord , which is the expression of radii of two circles and angle btw them ,, put thetha = 90 degree and use pythagoras theorem to get the result ..

Rudraksh Sisodia - 4 years, 9 months ago

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