A B is a chord of a parabola y2=4 a x,(a0) with vertex A . B C is drawn perpendicular to A B meeting the axis at C. The projection of B C on the axis of the parabola is

Question

A B is a chord of a parabola y2=4 a x,(a>0) with vertex A . B C is drawn perpendicular to A B meeting the axis at C. The projection of B C on the axis of the parabola is (A) a unit (B) 2 a unit (C) 8a unit (D) 4a unit

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<img class="img-fluid question-image" alt="image" src="https://cdn.tardigrade.in/img/question/mathematics/c1dad47d83f0c5fce25f2c0df57b0d27-.png" /> B(a t2, 2 a t): D(a t2, 0) mA B=(2 a t/a t2)=(2/t): mB C=-(t/2) .∴ y-2 a t=-(t/2)(x-a t2) text [Equation of B C] text for y=0,-2 a t=-(t/2)(x-a t2)=-(t/2) x+(a t3/2) ⇒ t x=4 a t+a t3 ⇒ x=4 a+a t2 ∴ c(4 a+a t2, 0) ∴ text Projection of B C text on the axis is D C=A C-A D=4 a+a t2-a t2=4 a

Q. AB is a chord of a parabola y2=4ax,(a>0) with vertex A.BC is drawn perpendicular to AB meeting the axis at C. The projection of BC on the axis of the parabola is

a unit

2 a unit

8a unit

4a unit

B(at2,2at):D(at2,0) mAB​=at22at​=t2​:mBC​=−2t​ ∴y−2at=−2t​(x−at2) [Equation of BC] for y=0,−2at=−2t​(x−at2)=−2t​x+2at3​ ⇒tx=4at+at3 ⇒x=4a+at2 ∴c(4a+at2,0) ∴ Projection of BC on the axis is DC=AC−AD=4a+at2−at2=4a

Q. $A B$ is a chord of a parabola $y^{2} = 4 a x, (a > 0)$ with vertex $A . BC$ is drawn perpendicular to $A B$ meeting the axis at $C$ . The projection of $BC$ on the axis of the parabola is