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The centres of those circles which touch the circle, x2 + y2 – 8x – 8y – 4 = 0, externally and also touch the x-axis,lie on: [JEE M 2016]
- a)a hyperbola
- b)a parabola
- c)a circle
- d)an ellipse which is not a circle
Correct answer is option 'B'. Can you explain this answer?
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Verified Answer
The centres of those circles which touch the circle, x2 + y2 – 8...
For the given circle, centre : (4, 4) radius = 6
(h – 4)2 = 20k + 20
∴ locus of (h, k) is (x – 4)2 = 20(y + 1), which is a parabola.
∴ locus of (h, k) is (x – 4)2 = 20(y + 1), which is a parabola.
Most Upvoted Answer
The centres of those circles which touch the circle, x2 + y2 – 8...
The centers of the circles that touch the circle with equation x^2 + y^2 = r^2 can be found by using the formula for the distance between two points.
Let's assume the circle with equation x^2 + y^2 = r^2 is centered at the origin (0,0). The distance between the origin and a point (x, y) on the circle is given by the formula:
d = sqrt((x - 0)^2 + (y - 0)^2)
= sqrt(x^2 + y^2)
Now, let's consider a circle with center (a, b) that touches the circle at a single point. The distance between the centers of the two circles is equal to the sum of their radii. Therefore, we have:
sqrt((a - 0)^2 + (b - 0)^2) = r1 + r2
Simplifying the equation, we get:
sqrt(a^2 + b^2) = r1 + r2
Now, we can solve for the centers (a, b) of the circles that touch the circle x^2 + y^2 = r^2 by finding all possible values of (a, b) that satisfy the above equation.
Let's assume the circle with equation x^2 + y^2 = r^2 is centered at the origin (0,0). The distance between the origin and a point (x, y) on the circle is given by the formula:
d = sqrt((x - 0)^2 + (y - 0)^2)
= sqrt(x^2 + y^2)
Now, let's consider a circle with center (a, b) that touches the circle at a single point. The distance between the centers of the two circles is equal to the sum of their radii. Therefore, we have:
sqrt((a - 0)^2 + (b - 0)^2) = r1 + r2
Simplifying the equation, we get:
sqrt(a^2 + b^2) = r1 + r2
Now, we can solve for the centers (a, b) of the circles that touch the circle x^2 + y^2 = r^2 by finding all possible values of (a, b) that satisfy the above equation.
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The centres of those circles which touch the circle, x2 + y2 – 8x – 8y – 4 = 0, externally and also touch the x-axis,lie on: [JEE M 2016]a)a hyperbolab)a parabolac)a circled)an ellipse which is not a circleCorrect answer is option 'B'. Can you explain this answer?
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