Parabola Tangent: GP Relation for Fixed Point Chords

In summary, the problem asks for a chord of contact between a parabola and a circle that passes through a fixed point, and it's not clear which tangent line should be used.
  • #1
DaalChawal
87
0
Tangent is drawn at any point ( $x_1$ , $y_1$ ) other than vertex on the parabola $y^2$ = 4ax . If tangents are drawn from any point on this tangent to the circle $x^2$ + $y^2$ = $a^2$ such that all chords of contact pass through a fixed point ( $x_2$ , $y_2$ ) then
(A) $x_1$ , a , $x_2$ are in G.P.
(B) $y_{1} \over 2$ ,a, $y_2$ are in G.P.
(C) -4 , $y_{1} \over y_{2}$ , $x_{1} \over x_{2}$ are in G.P.
(D) $x_1$ $x_2$ + $y_1$ $y_2$ = $a^2$
 
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  • #2
DaalChawal said:
Tangent is drawn at any point $( x1,y_1 )$ other than vertex on the parabola $y^2 = 4ax$.
let $(x_1,y_1) = (a, 2a)$

slope of the tangent line at $(a,2a)$ is $m = 1$

tangent line equation is $y - 2a = x-a \implies y = x+a$

$y = x+a$ intersects the circle $x^2+y^2 = a^2$ when $x^2 + (x+a)^2 = a^2 \implies 2x(x+a) = 0 \implies x_2 = 0 \text{ or } x_2 = -a$

$x_2 = 0 \implies y_2 = a$

(A) $\{x_1, a, x_2\} = \{a, a, 0\}$

(B) $\{y_1/2, a, y_2\} = \{a, a, a\}$

(C) $\{-4, y_1/y_2, x_1/x_2 \} = \{-4, 2, \emptyset \}$

(D) $x_1x_2 + y_1y_2 = a \cdot 0 + 2a \cdot a = 2a^2$now, check the four choices for $x_2=-a$
 
  • #3
DaalChawal said:
If tangents are drawn from any point on this tangent to the circle x2x2x^2 + y2y2y^2 = a2a2a^2
skeeter said:
y=x+ay=x+ay = x+a intersects the circle x2+y2=a2x2+y2=a2x^2+y^2 = a^2
Here the question says that from any point on the tangent to parabola, a tangent to the circle is drawn and that tangent is the chord of contact of parabola which passes through ( $x_2$ , $y_2$ ). And you have taken that tangent of parabola as chord of contact of circle also question says ( $x_2$ , $y_2$ ) is a fixed point that does not mean it lies on circle. Do correct me if I'm wrong.
 
  • #4
So, are there two different tangent lines, one tangent to the parabola and the other tangent to the circle?

Has this problem been translated to English?

Maybe you can post your interpretation with a sketch?
 
  • #5
Yes, there are two different tangent lines.
skeeter said:
Has this problem been translated to English?

No, you can look it's the question no. 255
photo_2021-04-06_21-09-17.jpg


This is what I'm saying

WIN_20210410_00_19_18_Pro.jpg
 

Related to Parabola Tangent: GP Relation for Fixed Point Chords

1. What is a parabola tangent?

A parabola tangent is a line that touches a parabola at only one point, known as the point of tangency. It is perpendicular to the parabola's axis of symmetry at that point.

2. How is the GP relation used to find the tangent of a parabola?

The GP (geometric progression) relation is used to find the tangent of a parabola by using the fixed point chords method. This involves selecting two points on the parabola, drawing a chord between them, and then finding the midpoint of the chord. The tangent line will pass through this midpoint and the point of tangency.

3. What is the significance of fixed point chords in relation to parabola tangents?

Fixed point chords are important in finding parabola tangents because they provide a way to determine the slope of the tangent line at a given point. By using the GP relation and fixed point chords, we can find the slope of the tangent line without needing to know the equation of the parabola.

4. Can the GP relation be used for any parabola?

Yes, the GP relation can be used for any parabola, regardless of its orientation or position on the coordinate plane. As long as two points on the parabola can be selected and a chord can be drawn between them, the GP relation can be used to find the tangent line.

5. How is the GP relation related to the derivative of a parabola?

The GP relation is closely related to the derivative of a parabola. In fact, the GP relation can be used to find the derivative of a parabola at a specific point. The slope of the tangent line at that point is equal to the derivative of the parabola at that point.

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