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The tangent at the point P (x1,y1)to the parabola y2=4ax meets the parabola y2=4a(x+b) at Q and R. the coordinates of the midpoint of QR are
A. (x1+b,y1+b)
B. (x1b,y1b)
C. (x1,y1)
D. (x1+b,y1)

Answer
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Hint:
Here we plot the graph of two parabolas to get a rough idea about the question.
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Complete step by step solution:
The equation of the tangent of parabola y2=4ax at any point P (h,k) is given by ky=2a(x+h)
Let (p,q)be the midpoint of the chord
Given, the tangent at the point P (x1,y1) to the parabola y2=4axmeets the parabola y2=4a(x+b)at Q and R.
Find the equation of the tangent of the parabola by substituting y1for kand x1 for h in ky=2a(x+h).
Equation of the tangent is given by yy1=2a(x+x1).
yy12ax2ax1=0(1)
The equation of chord with midpoint (p,q)is given by qy2a(x+p)4ab=q24a(q+b).
2ax+qy+2ahq2=0(2)
The tangent and the chord are the same line therefore equation (1) and (2) are the same. Compare both the equations.
2a2a=y1q=2ax12apq2
This implies that y1=q and 2ax1=q22ap
Substitute y1for q
2ax1=y122ap
Substitute 4ax1 for y12 and solve to obtain the value of p
2ax1=4ax12apx1=p

Therefore, the midpoint of QR is (x1,y1).

Therefore, Option C is correct.

Note:
In these types of questions, the knowledge of the equation of tangent and chord is very important. And the substitution of points should be done carefully. Students should know equation of parabola y2=4ax has centre (0,0) and equation of parabola y2=4a(x+b) has centre (b,0).
* Students are recommended to map the graph for better understanding of the question as it helps to find out the tangent, slope of the tangent, chord, normal and slope of the normal.
* A tangent is a straight line that just touches a point on a curve but never cuts the curve.
Equation of tangent of a parabola y2=4ax is given by y1y=2a(x+x1).
* A chord is a straight line segment joining any two points on a curve.
Equation of chord with midpoint (p,q) is given by qy2a(x+p)4ab=q24a(q+b).
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