Question

The tangent at the point P(x1y1) to the parabola y2 = 4ax meets the parabola y2 = 4a(x + b) at Q and R, the coordinates of the mid-point of QR are  

Solution

Correct option is

(x1y1)

Equation of the tangent at P(x1y1) to the parabola y2 = 4ax is

                          yy1 = 2a(x + x1)

or                     2ax – y1y + 2ax1 = 0                      (1)

If M(h, k) is the mid-point of QR, then equation of QR a chord of the parabola y2 = 4a(x + b) in terms of its mid-point is

ky – 2a(x +h) – 4ab = k2 – 4a(h + b)

(using T = S’) or 2ax – ky + k2 – 2ah = 0              (2)

Since (1) and (2) represent the same line, we have

             

⇒            k = y1 and k2 – 2ah = 2ax1

⇒   y12 – 2ah = 2ax1­   ⇒ 4ax1 – 2ax­1 = 2ah

⇒                 h = x1

SIMILAR QUESTIONS

Q1

If Land L2 are the length of the segments of any focal chord of the parabola y2 = x, then  is equal to

Q2

The tangents at three points A, B, C on the parabola y2 = 4x, taken in pairs intersect at the points Pand R. If  be the areas of the triangles ABC and PQR respectively, then 

Q3

The locus of the mid-point of the line segment joining the focus to a moving point on the parabola y2 = 4ax is another parabola with directrix

Q4

Equation of a common tangent to the curves y2 = 8x and xy = –1 is 

Q5

Consider a parabola y2 = 4ax, the length of focal chord is l and the length of the perpendicular from vertex to the chord is pthen

Q6

Tangent are drawn to a parabola from a point T. If P, Q are the points of constant then perpendicular distance from P, T and upon the tangent at the vertex of the parabola are in.

Q7

Chord of the parabola  which subtend right angle at vertex pass through

Q8

The locus of the vertex of the family of parabolas   is