If the normals to the parabola y 2=4 ax at the point P meets the curve again in Q and y 1, y 2 be the ordinates of P and the middle point of PQ respectively then y 1 y 2 is equal to

Question

If the normals to the parabola y 2=4 ax at the point P meets the curve again in Q and y 1, y 2 be the ordinates of P and the middle point of PQ respectively then y 1 y 2 is equal to (A) -a2 (B) -2 a2 (C) -4 a2 (D) -8 a 2

Tardigrade Admin · Accepted Answer

<img class="img-fluid question-image" alt="image" src="https://cdn.tardigrade.in/img/question/mathematics/343267efe3e0dff488eb62472c501945-.png" /> t 2=- t 1-(2/ t 1) ldots .(1) text also at 12= x 1 ; at 22= p ; 2 at 1= y 1 ; 2 at 2= q q + y 1=2 y 2 2 at 2+2 at 1=2 y 2 ⇒ 2 a ( t 1+ t 2)=2 y 2 -(4 a / t 1)=2 y 2 text from (1); putting t 1=( y 1/2 a ) ( t 1+ t 2=-(2/ t 1)) 2 y 2=-((4 a )(2 a )/ y 1) ∴ y 1 y 2=-4 a 2 ⇒( C )

Q. If the normals to the parabola $y^{2} = 4 a x$ at the point $P$ meets the curve again in $Q$ and $y_{1}, y_{2}$ be the ordinates of $P$ and the middle point of $PQ$ respectively then $y_{1} y_{2}$ is equal to

$- a^{2}$

$- 2 a^{2}$

$- 4 a^{2}$

$- 8 a^{2}$

Q. If the normals to the parabola y2=4ax at the point P meets the curve again in Q and y1​,y2​ be the ordinates of P and the middle point of PQ respectively then y1​y2​ is equal to

−a2

−2a2

−4a2

−8a2

t2​=−t1​−t1​2​….(1) also at12​=x1​;at22​=p;2at1​=y1​;2at2​=q q+y1​=2y2​ 2at2​+2at1​=2y2​⇒2a(t1​+t2​)=2y2​ −t1​4a​=2y2​ from (1); putting t1​=2ay1​​(t1​+t2​=−t1​2​) 2y2​=−y1​(4a)(2a)​∴y1​y2​=−4a2⇒(C)

Q. If the normals to the parabola $y^{2} = 4 a x$ at the point $P$ meets the curve again in $Q$ and $y_{1}, y_{2}$ be the ordinates of $P$ and the middle point of $PQ$ respectively then $y_{1} y_{2}$ is equal to

$- a^{2}$

$- 2 a^{2}$

$- 4 a^{2}$

$- 8 a^{2}$