If a line x+y=1 cut the parabola y2=4 a x in points A and B and normals drawn at A and B meet at C. The normal to the parabola from C other, than above two meet the parabola in D, then point D is

Question

If a line x+y=1 cut the parabola y2=4 a x in points A and B and normals drawn at A and B meet at C. The normal to the parabola from C other, than above two meet the parabola in D, then point D is (A) (a, a) (B) (2 a, 2 a) (C) (3 a, 3 a) (D) (4 a, 4 a)

Tardigrade Admin · Accepted Answer

<img class="img-fluid question-image" alt="image" src="https://cdn.tardigrade.in/img/question/mathematics/67046e825174c6be4a10fad467a4fec9-.png" /> A(x1, y1), B(x2, y2), C(x3, y3) y1+y2+y3=0 .....(1) y2=4 a x y2=4 a(1-y) y2+4 a y-4 a=0 y1+y2=-4 a .....(2) Using (2) in (1) y3=4 a D(4 a, 4 a)

Q. If a line $x + y = 1$ cut the parabola $y^{2} = 4 a x$ in points $A$ and $B$ and normals drawn at $A$ and $B$ meet at $C$ . The normal to the parabola from $C$ other, than above two meet the parabola in $D$ , then point $D$ is

$(a, a)$

$(2 a, 2 a)$

$(3 a, 3 a)$

$(4 a, 4 a)$

$A (x_{1}, y_{1}), B (x_{2}, y_{2}), C (x_{3}, y_{3})$
$y_{1} + y_{2} + y_{3} = 0.....$ (1)
$y^{2} = 4 a x$
$y^{2} = 4 a (1 - y)$
$y^{2} + 4 a y - 4 a = 0$
$y_{1} + y_{2} = - 4 a .....$ (2)
Using $(2)$ in $(1)$
$y_{3} = 4 a$
$D (4 a, 4 a)$

Q. If a line x+y=1 cut the parabola y2=4ax in points A and B and normals drawn at A and B meet at C. The normal to the parabola from C other, than above two meet the parabola in D, then point D is

(a,a)

(2a,2a)

(3a,3a)

(4a,4a)