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Test: Conic Sections (23 July) - JEE MCQ


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15 Questions MCQ Test - Test: Conic Sections (23 July)

Test: Conic Sections (23 July) for JEE 2024 is part of JEE preparation. The Test: Conic Sections (23 July) questions and answers have been prepared according to the JEE exam syllabus.The Test: Conic Sections (23 July) MCQs are made for JEE 2024 Exam. Find important definitions, questions, notes, meanings, examples, exercises, MCQs and online tests for Test: Conic Sections (23 July) below.
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Test: Conic Sections (23 July) - Question 1
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The length of the minor axis (along y-axis) of an ellipse in the standard form is . If this ellipse touches the line x + 6y = 8; then its eccentricity is

Detailed Solution for Test: Conic Sections (23 July) - Question 1

2b = 
The equation of tangent to ellipse  = 1 is:
y = mx  , where m = 
So, the equation of tangent is:
y = 
But x + 6y = 8 is given to be a tangent.
So, on comparing, we get a = 4.
Now, e = 

Test: Conic Sections (23 July) - Question 2
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The locus of mid-points of the line segments joining (-3, -5) and the points on the ellipse  is:

Detailed Solution for Test: Conic Sections (23 July) - Question 2

General point on , is (2cos, 3sin)
Given: B(-3, -5)
Mid-point C: 
h = ; k = 
= 1
 36x2 + 16y2 + 108x + 80y + 145 = 0

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Test: Conic Sections (23 July) - Question 3
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The locus of the mid-point of the line segment joining the point (4, 3) and the points on the ellipse x2 + 2y2 = 4 is an ellipse with eccentricity:

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Test: Conic Sections (23 July) - Question 4
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Let the eccentricity of an ellipse  = 1, a > b, be . If this ellipse passes through the point , then a2 + b2 is equal to:
Detailed Solution for Test: Conic Sections (23 July) - Question 4



a2 + b2 = 16 + 15 = 31

Test: Conic Sections (23 July) - Question 5
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If 3x + 4y = 12is a tangent to the ellipse  = 1 for some a  R, then the distance between the foci of the ellipse is
Detailed Solution for Test: Conic Sections (23 July) - Question 5

A line y = mx + c is a tangent to the ellipse  = 1, if c2 = a2m2 + b2.
Here, equation of tangent: 4y = -3x + 12
 y = -x + 
 (3)2 = a2 + 9
 a2 = 9 ×  = 16
 Eccentricity of ellipse, e = 
 Distance between the foci = 2ae = 2 × 4 × 
= 2√7

Test: Conic Sections (23 July) - Question 6
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Which of the following points lies on the locus of the foot of perpendicular drawn upon any tangent to the ellipse,  from any of its foci?
Detailed Solution for Test: Conic Sections (23 July) - Question 6

Let the foot of perpendicular be (h, k).




This line passes through (h, k).
(k - mh)2 = 4m2 + 2 …(1)
Line perpendicular to tangent will have slope (-1/m).

my = -x + 
(h + mk)2 = 2 …(2)
Adding equations (1) and (2):
k2(1 + m2) + h2(1 + m2) = 4(1 + m2)
h2 + k2 = 4
x2 + y2 = 4 (Auxiliary circle)
(-1, ) lies on the locus.

*Answer can only contain numeric values
Test: Conic Sections (23 July) - Question 7
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Let the eccentricity of the hyperbola  be . If the equation of the normal at the point  on the hyperbola is x + y = , then  is equal to _____. (in integer)

Detailed Solution for Test: Conic Sections (23 July) - Question 7

*Answer can only contain numeric values
Test: Conic Sections (23 July) - Question 8
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Let the common tangents to the curves 4(x2 + y2) = 9 and y2 = 4x intersect at the point Q. Let an ellipse, centered at the origin O, have lengths of semi-minor and semi-major axes equal to OQ and 6, respectively. If e and l respectively denote the eccentricity and the length of the latus rectum of this ellipse, then 1/e2 is equal to ________. (in integer)

Detailed Solution for Test: Conic Sections (23 July) - Question 8


Test: Conic Sections (23 July) - Question 9
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If the ellipse  = 1 meets the line  = 1 on the x-axis and the line  = 1 on the y-axis, then the eccentricity of the ellipse is
Detailed Solution for Test: Conic Sections (23 July) - Question 9

Test: Conic Sections (23 July) - Question 10
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Let the foci of the ellipse  = 1 and the hyperbola  coincide. Then the length of the latus rectum of the hyperbola is:
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Test: Conic Sections (23 July) - Question 11
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The locus of the midpoints of the chord of the circle x2 + y2 = 25 which is tangent to the hyperbola  = 1 is:
Detailed Solution for Test: Conic Sections (23 July) - Question 11



Equation of chord,
y - k = -(x - h)
ky - k2 = -hx + h2
hx + ky = h2 + k2

Tangent to: 
c2 = a2m2 - b2

(x2 + y2)2 = 9x2 - 16y2

*Answer can only contain numeric values
Test: Conic Sections (23 July) - Question 12
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Let a line L1 be tangent to the hyperbola  = 1 and let L2 be the line passing through the origin and perpendicular to L1. If the locus of the point of intersection of L1 and L2 is (x2 + y2)2 = αx2 + βy2, then α + β is equal to _____.

Detailed Solution for Test: Conic Sections (23 July) - Question 12

The equation of L1 is:
 ...(i)
The equation of line L2 is:
 ...(ii)
∵ The required point of intersection of L1 and L2 is (x1, y1).
Then,
 ...(iii)
and  ...(iv)
From equations (iii) and (iv),
secθ =  and tanθ = 
∴ The required locus of (x1, y1) is:
(x2 + y2)2 = 16x2 - 4y2
∴ α = 16, β = -4
∴ α + β = 12

Test: Conic Sections (23 July) - Question 13
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Let P(3, 3) be a point on the hyperbola,  = 1. If the normal to it at P intersects the x-axis at (9, 0) and e is its eccentricity, then the ordered pair (a2, e2) is equal to:
Detailed Solution for Test: Conic Sections (23 July) - Question 13

Test: Conic Sections (23 July) - Question 14
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The point P (-2 , ) lies on the hyerpbola = 1 having eccentricity . If the tangent and normal at P to the hyerpbola intersect its conjugate axis at the points Q and R respectively, then QR is equal to:
Detailed Solution for Test: Conic Sections (23 July) - Question 14

Test: Conic Sections (23 July) - Question 15
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If the line x - 1 = 0 is a directrix of the hyperbola kx2 - y2 = 6, then the hyperbola passes through the point
Detailed Solution for Test: Conic Sections (23 July) - Question 15


By checking options,satisfies the above equation.

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