Find The Locus Of A Pint P which Moves Such That Two Of The Three Normal’s Drawn From It To The Parabola y2 = 4ax are Mutually Perpendicular.

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Question

Find the locus of a pint P which moves such that two of the three normal’s drawn from it to the parabola y2 = 4ax are mutually perpendicular.

Solution

Correct option is

y2 = a(x – 3a).

Let point P(h, k) at normal to parabola be y = mx – 2am – am3. If this normal passes through (h, k), then

am3 + (2a – h)m + k = 0                       … (1)

If m1, m2, m3 be the roots of (1), then

m1 + m2 + m3 = 0                                 … (2)

     … (3)

                                 … (4)

Here is given that m1 m2 = –1

So,   

Putting the value in (2) we get, .

 k2 = a(h – 3a)

So, locus of P is y2 = a(x – 3a).

Testing

SIMILAR QUESTIONS

Q1

t1’ and ‘t2’ are two points on the parabola y2 = 4x. If the chord joining them is a normal to the parabola at ‘t1’ then

Q2

The vertex of the parabola y2 = 8x is at the center of a circle and the parabola cuts the circle at the ends of its latus rectum. Then the equation of the circle is

Q3

Find the equation of the parabola whose focus is (1, 1) and the directrix is x + y + 1 = 0.

Q4

If the line 2x + 3y = 1 touch the parabola y2 = 4ax at the pointP. Find the focal distance of the point P.

Q5

Find the angle between the tangents of the parabola y2 = 8x, which are drawn from the point (2, 5).

Q6

Find the locus of middle point of chord y2 = 4ax drawn through vertex.

Q7

Find the locus of the mid-point of the chords of the parabola y2 = 4ax which subtend a right angle at the vertex of the parabola.

Q8

Show that the normal at a point (at2, 2at) on the parabola y2 = 2ax cuts the curve again at the point whose parameter .

Q9

Show that the normal at a point (at2, 2at) on the parabola y2 = 2ax cuts the curve again at the point whose parameter .

Q10

If normal at the point (at2, 2at) in the parabola y2 = 4axintersects the parabola again at the (am2, 2am), then find the minimum value of m2.