Question
Question: The focal chord to \({y^2} = 16x\) is tangent to \({\left( {x - 6} \right)^2} + {y^2} = 2\), then th...
The focal chord to y2=16x is tangent to (x−6)2+y2=2, then the possible value of the slope of this chord are
A). (−1,1)
B). (−2,2)
C). (−2,21)
D). (2,−21)
Solution
Hint: Here, we will proceed by comparing the given equations with general equations of parabola and circle.
The given equation of the parabola is y2=16x →(1)
The general form of an upward parabola having focus F(a,0) is y2=4ax →(2)
On comparing above equations (1) and (2), we get
4a=16⇒a=4
Hence, focus of the given parabola is F(4,0)
It is also given that the focal chord of the given parabola is tangent to the circle (x−6)2+y2=2⇒(x−6)2+(y−0)2=(2)2 →(3).
Since, the general equation of a circle with centre coordinate as C(x1,y1) and radius as r is given by (x−x1)2+(y−y1)2=r2 →(4)
On comparing equations (3) and (4), we get
The centre coordinate of the given circle is C(6,0) and radius is 2.
Since we know that distance between two points A(a,b) and B(c,d) is given by distance formula which is d=(c−a)2+(d−b)2.
From the figure,
Distance between two points F(4,0) and C(6,0) is FC=(6−4)2+(0−0)2=(2)2=2
In the figure, we can see that △FCT is a right angled triangle with FC as hypotenuse.
Using Pythagoras theorem, we can write
(FC)2=(FT)2+(TC)2⇒(2)2=(FT)2+(r)2⇒(FT)2=4−r2=4−(2)2=4−2=2 ⇒FT=2
∴ tanθ=BasePerpendicular=FTTC=2r=22=1=tan45∘ ⇒θ=45∘
Since, line FC is a horizontal line.
Also we know that the slope of a line making an angle θ with the horizontal is given by m=tanθ
Therefore, the slope of the focal chord is m=tanθ=tan45∘=1
or m=tan(180∘−θ)=tan(180∘−45∘)=tan135∘=−1
Hence the slope of the required focal chord is (−1,1) which means option A is correct.
Note- In this particular problem line FC is a horizontal line because the y-coordinates of both the points (F and C) are equal. Here, if a line is making an angle θ with respect to the positive x-axis then the angle made by that line with the negative x-axis will be (180∘−θ). Slope can be measured either with respect to positive x-axis or with respect to negative x-axis.