Through the focus of the parabola y2 = 2px (p 0) a line is drawn which intersects the curve at A(x1 , y1 ) and B(x2 , y2 ). The ratio (y1y2/x1x2) equals

Question

Through the focus of the parabola y2 = 2px (p > 0) a line is drawn which intersects the curve at A(x1 , y1 ) and B(x2 , y2 ). The ratio (y1y2/x1x2) equals (A) 2 (B) - 1 (C) - 4 (D) some function of p

Tardigrade Admin · Accepted Answer

<img class="img-fluid question-image" alt="image" src="https://cdn.tardigrade.in/img/question/mathematics/43ecb7b4db2cc4025301edb5fb346bbc-.png" /> y 2 = 4ax, 4a = 2p > 0 x1 = at12 , y1 = 2at1 x2 = at 22, y2 = 2at2 and t1 t2 = - 1 ratio =(4a2t1t2/a2 t12t22) = -4 Note: for objective take focal chord as latus rectum

Q. Through the focus of the parabola $y^{2} = 2 p x (p > 0)$ a line is drawn which intersects the curve at $A (x_{1}, y_{1})$ and $B (x_{2}, y_{2})$ . The ratio $\frac{y _{1} y _{2}}{x _{1} x _{2}}$ equals

2

- 1

- 4

some function of p

$y^{2} = 4 a x, 4 a = 2 p > 0$
$x_{1} = a t_{1}^{2}, y_{1} = 2 a t_{1}$
$x_{2} = a t_{2}^{2}, y_{2} = 2 a t_{2}$
and $t_{1} t_{2} = - 1$
ratio $= \frac{4 a ^{2} t _{1} t _{2}}{a ^{2} t _{1}^{2} t _{2}^{2}} = - 4$
Note: for objective take focal chord as latus rectum

Q. Through the focus of the parabola y2=2px(p>0) a line is drawn which intersects the curve at A(x1​,y1​) and B(x2​,y2​). The ratio x1​x2​y1​y2​​ equals

2

- 1

- 4

some function of p

y2=4ax,4a=2p>0 x1​=at12​,y1​=2at1​ x2​=at22​,y2​=2at2​ and t1​t2​=−1 ratio =a2t12​t22​4a2t1​t2​​=−4 Note: for objective take focal chord as latus rectum

Q. Through the focus of the parabola $y^{2} = 2 p x (p > 0)$ a line is drawn which intersects the curve at $A (x_{1}, y_{1})$ and $B (x_{2}, y_{2})$ . The ratio $\frac{y _{1} y _{2}}{x _{1} x _{2}}$ equals

$y^{2} = 4 a x, 4 a = 2 p > 0$
$x_{1} = a t_{1}^{2}, y_{1} = 2 a t_{1}$
$x_{2} = a t_{2}^{2}, y_{2} = 2 a t_{2}$
and $t_{1} t_{2} = - 1$
ratio $= \frac{4 a ^{2} t _{1} t _{2}}{a ^{2} t _{1}^{2} t _{2}^{2}} = - 4$
Note: for objective take focal chord as latus rectum