Question
Question: Let \( {S_1} = 0 \) and \( {S_2} = 0 \) be two circles intersecting at \( P(6,4) \) and both are tan...
Let S1=0 and S2=0 be two circles intersecting at P(6,4) and both are tangent to x-axis and line y=mx (where m>0 ). If the product of radii of the circles S1=0 and S2=0 is 352, then find the value of m.
Solution
Hint : To solve this question we will assume the coordinates of the centre of the circle and the radius of the circle then will find the angle of the trigonometric function and then the slope of the equation.
Complete step-by-step answer :
Let the centre of the circle be (h,r) as it touches the x-axis.
Therefore, the equation of circle is –
(x−h)2+(y−r)2=r2 .... (A)
The circle also touches at y=mx
Let m be the slope which can be expressed as –
m=tanθ .... (B) (Where the angle of the function is made with x-axis)
∴r=htan(2θ)
Tangent is the inverse function of cot. So, place tan(2θ)=cot(2θ)1 in the above equation.
∴r=hcot(2θ)1
Do cross multiplication, Where numerator of one side is multiplied with the denominator of the opposite side.
⇒r.cot(2θ)=h
The above equation can be re-written as –
⇒h=r.cot(2θ) .... (C)
Place the above value in equation (A)
(x−r.cot(2θ))2+(y−r)2=r2
Since both the circles passes through the points (6,4)
⇒(6−r.cot(2θ))2+(4−r)2=r2
Simplify the above equation using the identity (a−b)2=a2−2ab+b2
⇒36−r2.cot2(2θ)−12r.cot(2θ)+16r2−8r=r2
Simplify the above equation –
⇒r2.cot2(2θ)+r(−12r.cot(2θ)−8)+52=0
Product of roots is given by 352
We know that in quadratic equation –
Product of roots is given by ac
Now,
ac=352
Here we have a=cot2(2θ) and c=52 place values in the above equation –
⇒cot2(2θ)52=352
Common multiples from both the sides of the equation cancel each other.
⇒cot2(2θ)1=31
The above equation implies –
⇒cot2(2θ)=3
Cot is the inverse function of tangent –
⇒tan2(2θ)1=3
Cross-multiplication implies –
⇒tan2(2θ)=31
Take square-root on both the side of the equation –
⇒tan(2θ)=±31
Since the circle is in the first quadrant, the tangent is positive.
⇒tan(2θ)=31
By referring to the trigonometric table for values –
⇒tan(2θ)=tan6π
Tangent on both the sides of the equation cancels each other.
⇒(2θ)=6π
Simplify the above equation-
⇒θ=6π×2
Common factors from the numerator and the denominator cancel each other.
⇒θ=3π
Place the above value in equation (B)
tanθ=3 ⇒m=3
This is the required solution.
So, the correct answer is “ m=3 ”.
Note : Follow the All STC rule, and first check for which quadrant the point lies and then take positive or the negative signs for the trigonometric functions. It is most important to follow the correct sign convention.