Find the equation of the parabola if the curve is open leftward, vertex is (2, 1) and passing through the poin

Question 103 of 126

Q. Find the equation of the parabola if the curve is open leftward, vertex is (2, 1) and passing through the point (1, 3).

Puducherry TnboardTamil Nadu HSC (DGE) Board 2020Subjective· 2mImportance★★★★★

82% · 103/126 Questions

🔒 Locked · start free trial →

You're viewing a preview — the full solution, concept, methods & PYQ mapping are locked.

Start your 14-day free trial to unlock the full solution →

Concept understanding — Parabola — Standard Equation

Setting eccentricity $e=1$ in the conic definition ( $SP=e\cdot PM$ ) gives the parabola — the locus of points equidistant from a fixed focus and a fixed directrix.

Standard form (vertex at the origin, focus on the axis). With focus $(a,0)$ and directrix $x=-a$ : $y^2=4ax$ (opens right). The other three orientations, by symmetry: $y^2=-4ax$ (left), $x^2=4ay$ (up), $x^2=-4ay$ (down).

Vocabulary. Axis: line through the focus $\perp$ directrix. Vertex: where the axis meets the curve. Focal chord: any chord through the focus. Latus rectum: the focal chord $\perp$ the axis; for $y^2=4ax$ its endpoints are $(a,\pm2a)$ , so its length is $4a$ .

Vertex at $(h,k)$ :

Equation	Focus	Directrix
$(y-k)^2=4a(x-h)$	$(h+a,k)$	$x=h-a$
$(y-k)^2=-4a(x-h)$	$(h-a,k)$	$x=h+a$
$(x-h)^2=4a(y-k)$	$(h,k+a)$	$y=k-a$
$(x-h)^2=-4a(y-k)$	$(h,k-a)$	$y=k+a$

Start your 14-day free trial to unlock the full solution →

Sets up the standard leftward-opening parabola form at the given vertex, then solves for the parameter $a$ using the given point on the curve.

A parabola that opens leftward with vertex $(h,k)$ has the standard equation $(y-k)^2=-4a(x-h)$ , $a>0$ .
Here vertex $(h,k)=(2,1)$ , so the equation is $(y-1)^2=-4a(x-2)$ .
The curve passes through $(1,3)$ , so substitute $x=1,\ y=3$ : $(3-1)^2=-4a(1-2)$ .
Simplify: $4=-4a(-1)=4a$ , so $a=1$ . …

Start your 14-day free trial to unlock the full solution →

←Previous questionQ33 — The radius of the circle 3x² + by² + 4bx - 6by + b² = 0 is : (a) √(11)…→Next questionQ35 — A concrete bridge is designed as a parabolic arch. The road over bridg…

Q.Find the equation of the parabola if the curve is open leftward, vertex is (2, 1) and passing through the point (1, 3).

🎓Unlock everything free for 14 days

Unlock everything free for 14 days