A triangle PQR is inscribed in the parabola $$x2=8y$$ such that ∠$$Q=$$π$$2$$ . Let $$L1$$,$$L2be$$ the tangents drawn at P, R respectively. If the equation of the parabola whose axis is along $$y-axis$$, whose latus rectum is along $$x-axis$$ and its length is the minimum length of intercept on $$x-axis$$ cut off by $$L1$$,$$L2$$ is $$y=ax2+bx+c$$,then, which of the following $$is/are$$ true?

Question

A triangle PQR is inscribed in the parabola $$x2=8y$$ such that ∠$$Q=$$π$$2$$ . Let $$L1$$,$$L2be$$ the tangents drawn at P, R respectively. If the equation of the parabola whose axis is along $$y-axis$$, whose latus rectum is along $$x-axis$$ and its length is the minimum length of intercept on $$x-axis$$ cut off by $$L1$$,$$L2$$  is   $$y=ax2+bx+c$$,then, which of the following $$is/are$$ true?

Infinity Learn · Accepted Answer

$$P=4t1$$,$$2t12$$,$$Q=4t2$$,$$2t22$$,$$R=4t3$$,$$2t32PQ$$⊥$$QR=t1+t2t2+t3=$$−$$4$$⇒$$t22+t1+t3t2+t1t3+4=0$$Δ≥$$0$$⇒$$t1+t32-4t1t2+4$$≥$$0$$⇒$$t1$$−$$t3$$≥$$4$$ Minimum length of intercept by $$L1$$,$$L2$$ on $$x-axis$$ $$=2t1$$−$$t3=8given$$ parabola is $$y=ax2+bx+c$$ , whose axis is $$y-axis$$ ,so $$b=0and$$ length of the latusrectum is $$=1a=$$ mininmum of $$L1L2=8$$⇒$$a=$$±$$18$$ and the parabola passing through the points $$4$$,$$0$$ and $$-4$$,$$0$$⇒$$c=$$±$$2$$ Parabola is $$y=x28$$−$$2($$ or $$)y=$$−$$x28+2$$

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