If the normals from any point to the parabola `y^2=4x` cut the line `x=2` at points whose ordinates are in AP, then prove that the slopes of tangents

Question

If the normals from any point to the parabola `y^2=4x` cut the line `x=2` at points whose ordinates are in AP, then prove that the slopes of tangents

If the normals from any point to the parabola `y^2=4x` cut the line `x=2` at points whose ordinates are in AP, then prove that the slopes of tangents at the co-normal points are in GP.

Monjur Alahi

9 views

2025-01-04 04:42

1 Answers

Related Questions

A point P moves such that sum of the slopes of the normals drawn from it to the hyperbola xy=16 is equal to the sum of ordinates of feet of normals. T

2 Answers 6 Views

If the lines `L_1a n dL_2` are tangents to `4x^2-4x-24 y+49=0` and are normals for `x^2+y^2=72 ,` then find the slopes of `L_1` and `L_2dot`

2 Answers 5 Views

Tangents are drawn to the parabola `y^2=4a x` at the point where the line `l x+m y+n=0` meets this parabola. Find the point of intersection of these t

2 Answers 5 Views

Three normals are drawn from the point (7, 14) to the parabola `x^2-8x-16 y=0` . Find the coordinates of the feet of the normals.

2 Answers 5 Views

`P(t_(1))andQ(t_(2))` are points `t_(1)andt_(2)` on the parabola `y^(2)=4ax`. The normals at P and Q meet on the parabola. Show that the middle point

2 Answers 4 Views

Double ordinate `A B` of the parabola `y^2=4a x` subtends an angle `pi/2` at the focus of the parabola. Then the tangents drawn to the parabola at `Aa

2 Answers 6 Views

`P ,Q ,` and `R` are the feet of the normals drawn to a parabola `(y-3)^2=8(x-2)` . A circle cuts the above parabola at points `P ,Q ,R ,a n dS` . The

2 Answers 5 Views

Normals at two points `(x_1y_1)a n d(x_2, y_2)` of the parabola `y^2=4x` meet again on the parabola, where `x_1+x_2=4.` Then `|y_1+y_2|` is equal to `

2 Answers 4 Views

In parabola y^2=4x, From the point (15,12), three normals are drawn to the three co-normals point is

2 Answers 6 Views

Points A, B, C lie on the parabola `y^2=4ax` The tangents to the parabola at A, B and C, taken in pair, intersect at points P, Q and R. Determine the

2 Answers 6 Views

Salim Ahmed Kawsar · Accepted Answer · 2023-02-05T15:29:48+06:00

The equation of the normal to the parabola `y^(2)=4` is given by
`y=-tx+2t+t^(3)` (1)
Since it intersects x=2, we get ` y=t^(3)`
Let the three ordinates `t_(1)^(3),t_(2)^(3), andt_(3)^(3)` be in AP. Then,
`2_(2)^(3)=t_(1)^(3)+t_(3)^(3)`
`=(t_(1)+t_(3))^(3)-3t_(1)t_(3)(t_(1)+t_(3))` (2)
Now, `t_(1)+t_(2)+t_(3)=0`
`or" "t_(1)+t_(3)=-t_(2)`
Hence, (2) reduces to
`2t_(2)^(3)=-(-t_(2))^(3)-3t_(1)t_(3)(-t_(2))`
`=-(t_(2)^(3)+3t_(1)t_(2)t)(3)`
`3t_(2)^(3)+3t_(1)t_(2)t_(3)ort_(2)^(2)=t_(1)t_(3)`
Therefore, `t_(1),t_(2)andt_(3)` are tangents `1//t_(1),1//t_(2),and1//t_(3)` are in GP.