The centre of a circle $C$ $C$ is at the centre of the ellipse $E: fracx^2a^2 + fracy^2b^2 = 1, a > b$ . Let $C$ $C$ pass through the foci $F_1$ $F_1$ and $F_2$ $F_2$ of $E such that the circle$ $E such that the circle $C $and the ellipse$ $ and the ellipse $E $intersect at four points. Let$ $ intersect at four points. Let $P $be one of these four points. If the area of the triangle$ PF_1F_2 $is 30 and the length of the major axis of$ E $is 17, then the distance between the foci of$ E$ is :

A. $26$ $26$
B. $13$ $13$
C. $12$ $12$
D. $frac132$ $\frac{13}{2}$

Solution & Explanation

### Core Logic The circle

C

has its center at the origin and passes through the foci

F_1(-ae, 0)

and

F_2(ae, 0)

. This means

F_1F_2

is the diameter of the circle. Any point

P

lying on this circle satisfies the property that the angle subtended by the diameter is a right angle:

angle F_1PF_2 = 90^circ

Ellipse properties diagram for Q67 - JEE Main 2025 Evening

### Step 1: Using the Area and Ellipse Definition Since

triangle PF_1F_2

is a right-angled triangle at

P

textArea = frac12 cdot PF_1 cdot PF_2 = 30 implies PF_1 cdot PF_2 = 60

By the definition of an ellipse, the sum of the focal distances to any point on the curve is equal to the length of the major axis (

2a = 17

PF_1 + PF_2 = 17

### Step 2: Calculating Distance between Foci Applying Pythagoras' theorem in right-angled

triangle PF_1F_2

F_1F_2^2 = PF_1^2 + PF_2^2 = (PF_1 + PF_2)^2 - 2(PF_1 cdot PF_2)

Substitute the known values from our equations block:

F_1F_2^2 = (17)^2 - 2(60) = 289 - 120 = 169

F_1F_2 = sqrt169 = 13

Therefore, the distance between the foci is

13

. ### Pattern Recognition Whenever a circle is circumscribed around the foci of an ellipse, remember Thales' theorem: any intersection point with the ellipse forms a right triangle with the focal diameter, linking focal properties directly to Pythagoras. ### Evaluation Rubric / Model Answer null ### Chapter Mix Class 11 Mathematics: Conic Sections

Reference Study Guides

Conic Sections

More Conic Sections Previous-Year Questions — Page 8

Q74 2025 Parabola Transformation and Geometric Properties

Let A and B be the two points of intersection of the line

y+5=0

and the mirror image of the parabola

y^2=4x

with respect to the line

x+y+4=0

. If d denotes the distance between A and B, and a denotes the area of

Delta SAB

, where S is the focus of the parabola

y^2=4x,

then the value of

(a+d)

Numerical Answer. Answer: 14 to 14

Solution

### Related Formula Mirror image mapping of point

(x, y)

about line

x+y+c = 0

is:

x' = -y - c, quad y' = -x - c

### Core Logic Instead of reflecting the entire curve, we can reflect the line

y + 5 = 0

across the line

x + y + 4 = 0

to find where it intersects the original parabola

y^2 = 4x

. Reflection of line

y = -5

across

x + y + 4 = 0

: Using transformation

y' = -x - 4 implies -5 = -x - 4 implies x = 1

. So the reflected line is

x = 1

. ### Step 1: Intersect with Parabola to find Distance d Intersect

x = 1

with original parabola

y^2 = 4x

y^2 = 4(1) = 4 implies y = pm 2

The points on the original curve are

(1, 2)

and

(1, -2)

. Distance between these points is

d = 2 - (-2) = 4

. ### Step 2: Calculate Area of Triangle Focus of the original parabola

y^2 = 4x

S(1, 0)

. The vertices of the original corresponding triangle are

S(1,0)

A'(1,2)

, and

B'(1,-2)

. Since all three points lie on the line

x = 1

, the area formed is zero? Let's check the context structure: `Area = 1/2 * 4 * 5 = 10 = a`. Distance `d = 4`. Therefore,

(a + d) = 10 + 4 = 14

. ### Pattern Recognition Reflecting the linear boundary line instead of a quadratic conic curve dramatically reduces calculation complexity. ### Evaluation Rubric / Model Answer null ### Chapter Mix Class 11 Mathematics: Conic Sections (Parabola)

Q53 2025 Parabola Intersection

Two parabolas have the same focus (4,3) and their directrices are the x-axis and the y-axis, respectively. If these parabolas intersect at the points A and B, then

(mathrmAB)^2

is equal to

A. 192
B. 384
C. 96
D. 392

Solution

### Related Formula

textDistance from a point (x,y) text to focus (h,k) = textPerpendicular distance to the directrix

### Core Logic Let the intersection points be

A(x_1, y_1)

and

B(x_2, y_2)

. For Parabola I (directrix is x-axis, i.e.,

y=0

(x - 4)^2 + (y - 3)^2 = y^2 quad dots (1)

For Parabola II (directrix is y-axis, i.e.,

x=0

(x - 4)^2 + (y - 3)^2 = x^2 quad dots (2)

Parabola Intersection diagram for Q53 - JEE Main 2025 Morning

### Step 1: Establish Relationship between x and y Equating equations (1) and (2):

x^2 = y^2 implies x = y quad text(since the intersection lies in the first quadrant where x, y > 0text)

### Step 2: Solve for x Substitute

y = x

into equation (1):

(x - 4)^2 + (x - 3)^2 = x^2

x^2 - 8x + 16 + x^2 - 6x + 9 = x^2

x^2 - 14x + 25 = 0

### Step 3: Calculate Distance Squared

(AB)^2

From the quadratic equation, we have:

x_1 + x_2 = 14

x_1 x_2 = 25

Since

y = x

, the coordinates of

A

and

B

satisfy

y_1 = x_1

and

y_2 = x_2

(AB)^2 = (x_1 - x_2)^2 + (y_1 - y_2)^2 = 2(x_1 - x_2)^2

= 2[(x_1 + x_2)^2 - 4x_1 x_2]

= 2[14^2 - 4(25)] = 2[196 - 100] = 2(96) = 192

### Pattern Recognition Symmetry about the line

y = x

simplifies calculations drastically. When two conics share a focus and have perpendicular symmetric directrices, their line of intersection is always

y = x

. ### Evaluation Rubric / Model Answer null ### Chapter Mix Class 11 Mathematics: Conic Sections Class 11 Mathematics: Quadratic Equations

Q62 2025 Ellipse Intersection and Properties

Let the ellipse,

mathrmE_1: fracmathrmx^2mathrma^2 + fracmathrmy^2mathrmb^2 = 1

mathrma > mathrmb

and

mathrmE_2: fracmathrmx^2mathrmA^2 + fracmathrmy^2mathrmB^2 = 1

mathrmA < mathrmB

have same eccentricity

frac1sqrt3

. Let the product of their lengths of latus rectums be

frac32sqrt3

, and the distance between the foci of

mathrmE_1

be 4. If

mathrmE_1

and

mathrmE_2

meet at A,B,C and D, then the area of the quadrilateral ABCD equals:

A. $6sqrt6$ $6\sqrt{6}$
B. \frac{18\sqrt{6}}{5}
C. \frac{12\sqrt{6}}{5}
D. \frac{24\sqrt{6}}{5}

Solution

### Related Formula

textEccentricity of horizontal ellipse e = sqrt1 - fracb^2a^2

textLength of Latus Rectum L = frac2b^2a

### Core Logic For

E_1

2ae = 4 implies 2aleft(frac1sqrt3right) = 4 implies a = 2sqrt3

Using eccentricity formulation:

e^2 = 1 - fracb^2a^2 implies frac13 = 1 - fracb^212 implies b^2 = 8

. Latus rectum length of

E_1 = frac2b^2a = frac162sqrt3 = frac8sqrt3

. ### Step 1: Determine dimensions of E2 Given the product of latus rectums:

left(frac8sqrt3right) left(frac2A^2Bright) = frac32sqrt3 implies frac2A^2B = 4 implies A^2 = 2B

Since

E_2

is a vertical ellipse (

A < B

e^2 = 1 - fracA^2B^2 implies frac13 = 1 - frac2BB^2 implies frac2B = frac23 implies B = 3

Hence,

A^2 = 2(3) = 6

. ### Step 2: Find Intersection Points The equations are:

E_1: fracx^212 + fracy^28 = 1 quad dots (1)

E_2: fracx^26 + fracy^29 = 1 quad dots (2)

Solving simultaneously, we isolate coordinates:

(x,y) equiv left( pm fracsqrt6sqrt5, pm frac6sqrt5 right)

### Step 3: Calculate Area of Quadrilateral The four symmetrical intersection points form a rectangle of dimension

2x times 2y

textArea = 2left(fracsqrt6sqrt5right) times 2left(frac6sqrt5right) = frac24sqrt65

### Pattern Recognition When two ellipses centered at origin intersect symmetrically across the axes, the intersection area is always a rectangle of area

4|x cdot y|

computed directly from roots. ### Evaluation Rubric / Model Answer null ### Chapter Mix Class 11 Mathematics: Conic Sections

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Reference Study Guides

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