1
JEE Main 2022 (Online) 27th June Evening Shift
+4
-1

If the equation of the parabola, whose vertex is at (5, 4) and the directrix is $$3x + y - 29 = 0$$, is $${x^2} + a{y^2} + bxy + cx + dy + k = 0$$, then $$a + b + c + d + k$$ is equal to

A
575
B
$$-$$575
C
576
D
$$-$$576
2
JEE Main 2022 (Online) 27th June Morning Shift
+4
-1

Let the eccentricity of an ellipse $${{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} = 1$$, $$a > b$$, be $${1 \over 4}$$. If this ellipse passes through the point $$\left( { - 4\sqrt {{2 \over 5}} ,3} \right)$$, then $${a^2} + {b^2}$$ is equal to :

A
29
B
31
C
32
D
34
3
JEE Main 2022 (Online) 26th June Evening Shift
+4
-1

If m is the slope of a common tangent to the curves $${{{x^2}} \over {16}} + {{{y^2}} \over 9} = 1$$ and $${x^2} + {y^2} = 12$$, then $$12{m^2}$$ is equal to :

A
6
B
9
C
10
D
12
4
JEE Main 2022 (Online) 26th June Evening Shift
+4
-1

The locus of the mid point of the line segment joining the point (4, 3) and the points on the ellipse $${x^2} + 2{y^2} = 4$$ is an ellipse with eccentricity :

A
$${{\sqrt 3 } \over 2}$$
B
$${1 \over {2\sqrt 2 }}$$
C
$${1 \over {\sqrt 2 }}$$
D
$${1 \over 2}$$
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