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Consider any point P on the ellipse x2/25 + y2/9= 1 in the first quadrant. Let r and s represent its distances from (4,0) and ( - 4 , 0 ) respectively, th en (r + s) is equal to
- a)10 unit
- b)9 unit
- c)8 unit
- d)6 unit
Correct answer is option 'A'. Can you explain this answer?
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Consider any point P on the ellipse x2/25 + y2/9= 1 in the first quadr...
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Consider any point P on the ellipse x2/25 + y2/9= 1 in the first quadr...
Explanation:
Given equation of the ellipse is x^2/25 + y^2/9 = 1.
Let P(x, y) be any point on the ellipse in the first quadrant.
We are given two points (4, 0) and (-4, 0) and we need to find the distances r and s of point P from these points.
Step 1: Find the coordinates of point P
Let's assume that the x-coordinate of P is a, where a > 0. Then, we can find the y-coordinate of P using the equation of the ellipse:
a^2/25 + y^2/9 = 1
Simplifying the equation, we get:
y^2 = 9 - (9/25)a^2
Taking the square root on both sides, we obtain:
y = ±sqrt(9 - (9/25)a^2)
Since P is in the first quadrant, y > 0. Therefore, we can write:
y = sqrt(9 - (9/25)a^2)
So, the coordinates of point P are (a, sqrt(9 - (9/25)a^2)).
Step 2: Find the distances r and s
The distance between two points (x1, y1) and (x2, y2) is given by the formula:
d = sqrt((x2 - x1)^2 + (y2 - y1)^2)
For point P and (4, 0), the distance r is given by:
r = sqrt((4 - a)^2 + (0 - sqrt(9 - (9/25)a^2))^2)
Simplifying the equation, we get:
r = sqrt((16 - 8a + a^2) + (9 - (9/25)a^2))
r = sqrt(25 - (8a - a^2) + 9 - (9/25)a^2)
r = sqrt(34 - (8a - a^2) - (9/25)a^2)
Similarly, for point P and (-4, 0), the distance s is given by:
s = sqrt((-4 - a)^2 + (0 - sqrt(9 - (9/25)a^2))^2)
Simplifying the equation, we get:
s = sqrt((16 + 8a + a^2) + (9 - (9/25)a^2))
s = sqrt(25 + (8a + a^2) + 9 - (9/25)a^2)
s = sqrt(34 + (8a + a^2) - (9/25)a^2)
Step 3: Find the sum of distances r and s
(r + s) = sqrt(34 - (8a - a^2) - (9/25)a^2) + sqrt(34 + (8a + a^2) - (9/25)a^2)
Simplifying the equation, we get:
(r + s) = sqrt(68 + 2(8a + a^2) - 2(9/25)a^2)
(r + s) = sqrt(68 + 16a + 2a^2 - (18/25)a^
Given equation of the ellipse is x^2/25 + y^2/9 = 1.
Let P(x, y) be any point on the ellipse in the first quadrant.
We are given two points (4, 0) and (-4, 0) and we need to find the distances r and s of point P from these points.
Step 1: Find the coordinates of point P
Let's assume that the x-coordinate of P is a, where a > 0. Then, we can find the y-coordinate of P using the equation of the ellipse:
a^2/25 + y^2/9 = 1
Simplifying the equation, we get:
y^2 = 9 - (9/25)a^2
Taking the square root on both sides, we obtain:
y = ±sqrt(9 - (9/25)a^2)
Since P is in the first quadrant, y > 0. Therefore, we can write:
y = sqrt(9 - (9/25)a^2)
So, the coordinates of point P are (a, sqrt(9 - (9/25)a^2)).
Step 2: Find the distances r and s
The distance between two points (x1, y1) and (x2, y2) is given by the formula:
d = sqrt((x2 - x1)^2 + (y2 - y1)^2)
For point P and (4, 0), the distance r is given by:
r = sqrt((4 - a)^2 + (0 - sqrt(9 - (9/25)a^2))^2)
Simplifying the equation, we get:
r = sqrt((16 - 8a + a^2) + (9 - (9/25)a^2))
r = sqrt(25 - (8a - a^2) + 9 - (9/25)a^2)
r = sqrt(34 - (8a - a^2) - (9/25)a^2)
Similarly, for point P and (-4, 0), the distance s is given by:
s = sqrt((-4 - a)^2 + (0 - sqrt(9 - (9/25)a^2))^2)
Simplifying the equation, we get:
s = sqrt((16 + 8a + a^2) + (9 - (9/25)a^2))
s = sqrt(25 + (8a + a^2) + 9 - (9/25)a^2)
s = sqrt(34 + (8a + a^2) - (9/25)a^2)
Step 3: Find the sum of distances r and s
(r + s) = sqrt(34 - (8a - a^2) - (9/25)a^2) + sqrt(34 + (8a + a^2) - (9/25)a^2)
Simplifying the equation, we get:
(r + s) = sqrt(68 + 2(8a + a^2) - 2(9/25)a^2)
(r + s) = sqrt(68 + 16a + 2a^2 - (18/25)a^
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Consider any point P on the ellipse x2/25 + y2/9= 1 in the first quadrant. Let r and s represent its distances from (4,0) and ( - 4 , 0 ) respectively, th en (r + s) is equal toa)10 unitb)9 unitc)8 unitd)6 unitCorrect answer is option 'A'. Can you explain this answer?
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Consider any point P on the ellipse x2/25 + y2/9= 1 in the first quadrant. Let r and s represent its distances from (4,0) and ( - 4 , 0 ) respectively, th en (r + s) is equal toa)10 unitb)9 unitc)8 unitd)6 unitCorrect answer is option 'A'. Can you explain this answer? for Defence 2024 is part of Defence preparation. The Question and answers have been prepared according to the Defence exam syllabus. Information about Consider any point P on the ellipse x2/25 + y2/9= 1 in the first quadrant. Let r and s represent its distances from (4,0) and ( - 4 , 0 ) respectively, th en (r + s) is equal toa)10 unitb)9 unitc)8 unitd)6 unitCorrect answer is option 'A'. Can you explain this answer? covers all topics & solutions for Defence 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Consider any point P on the ellipse x2/25 + y2/9= 1 in the first quadrant. Let r and s represent its distances from (4,0) and ( - 4 , 0 ) respectively, th en (r + s) is equal toa)10 unitb)9 unitc)8 unitd)6 unitCorrect answer is option 'A'. Can you explain this answer?.
Consider any point P on the ellipse x2/25 + y2/9= 1 in the first quadrant. Let r and s represent its distances from (4,0) and ( - 4 , 0 ) respectively, th en (r + s) is equal toa)10 unitb)9 unitc)8 unitd)6 unitCorrect answer is option 'A'. Can you explain this answer? for Defence 2024 is part of Defence preparation. The Question and answers have been prepared according to the Defence exam syllabus. Information about Consider any point P on the ellipse x2/25 + y2/9= 1 in the first quadrant. Let r and s represent its distances from (4,0) and ( - 4 , 0 ) respectively, th en (r + s) is equal toa)10 unitb)9 unitc)8 unitd)6 unitCorrect answer is option 'A'. Can you explain this answer? covers all topics & solutions for Defence 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Consider any point P on the ellipse x2/25 + y2/9= 1 in the first quadrant. Let r and s represent its distances from (4,0) and ( - 4 , 0 ) respectively, th en (r + s) is equal toa)10 unitb)9 unitc)8 unitd)6 unitCorrect answer is option 'A'. Can you explain this answer?.
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