JEE Maths›Conic Sections›Visual Solution

Visual SolutionHard

Question

Let

P

be a variable point on the ellipse

\frac{x^2}{25} + \frac{y^2}{16} = 1

with foci

S

and

S'

. If

A

is the area of triangle

PSS'

, then the maximum value of

A

is:

Solution Path

Base = distance between foci = 6, max height = semi-minor axis = 4, area = 12.

01Question Setup

1/4

Find the maximum area of triangle

PSS'

where

P

is on the ellipse

\frac{x^2}{25} + \frac{y^2}{16} = 1

and

S, S'

are foci.

Max area

= ;?

02Plot Ellipse

2/4

Ellipse with

a = 5

b = 4

c = 3

. Foci at

(\pm 3, 0)

. Base

SS' = 6

SS' = 6

03Maximum AreaKEY INSIGHT

3/4

Area

= \frac{1}{2} \times SS' \times h

. Height is maximized when

P

is at end of minor axis:

h = b = 4

Max height

= b = 4

04Final Answer

4/4

Area

= \frac{1}{2} \times 6 \times 4 = 12

\boxed{12}

Concepts from this question3 concepts unlocked

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Eccentricity and Conic Classification

EASY

The eccentricity e = c/a determines the type of conic: e < 1 gives an ellipse, e = 1 gives a parabola, and e > 1 gives a hyperbola. For a circle, e = 0.

e = \frac{c}{a}

First step in most conic problems. JEE frequently asks you to identify or compare conics based on eccentricity values.

Conic identificationEccentricity comparisonLocus problems

Practice (12 Qs) →

Focal Chord Property

STANDARD

For a focal chord with endpoints P and Q on a conic, the semi-latus rectum l satisfies 1/SP + 1/SQ = 2/l, where S is the focus. This is a harmonic mean relation between SP and SQ.

\frac{1}{SP} + \frac{1}{SQ} = \frac{2}{l}

Saves time in problems involving focal chords, latus rectum lengths, and minimum/maximum distance from the focus.

Focal chord lengthLatus rectumHarmonic mean problems

Practice (8 Qs) →

Parametric Representation of Conics

STANDARD

Parabola y² = 4ax uses (at², 2at). Ellipse x²/a² + y²/b² = 1 uses (a cos t, b sin t). Hyperbola x²/a² - y²/b² = 1 uses (a sec t, b tan t).

(at^2,\, 2at), \quad (a\cos t,\, b\sin t), \quad (a\sec t,\, b\tan t)

Parametric form simplifies tangent, normal, and chord equations. Most JEE solutions use parametric coordinates for cleaner algebra.

Tangent and normal equationsChord of contactArea of triangle on conic

Practice (10 Qs) →

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